How do you de-weed a very large overgrown garden?

3 inches of mulch would reduce the weeds dramatically.
A layer of plastic would allow weeds to sprout, but they would be suppressed underneath.
Cut holes in the plastic only where you plant and weeding will be confined only nearest the plants.
You could use chemicals that kill weeds but not your plants, but that would defeat the purpose of growing your own as the supermarket already has chemically laden food by the bushel.
Tilling the soil under a few times before planting helps reduce weeds. After you plant and your plants begin to grow, mulch (I use grass clippings) between the rows of your plants with a layer that will dry before applying the next.
A little hoeing (Just scraping the tops off) weeds is a quick way to eliminate them before they get too established.
Weeding is a work in progress and the more your weed (regularly) the less you'll have.
Some edible weeds like clover are beneficial and ok that they grow within your plantings. They are a good source plant to lock nitrogen within the soil and bees love them, then return the favor by pollinating.
Also of note: Only fertilize the hole(s) where you place each plant or seeds, with follow-up applications placed near the root base at appropriate intervals. Total cover fertilizing applications will "feed the weeds" and it's not recommended. Lime would be one exception to apply as a total cover. Most other fertilizers try to keep them near your plants.

Put down black plastic for a few weeks under hot sun. Or put down cardboard and build a raised bed. Either helps solve the weed problem. Once you have a garden put down mulch to squelch the weeds.

Some weed seeds can remain dormant in the soil up to ten years. You will always have some weeds germinating and coming up, but if you carefully tend the garden, there should be fewer weeds each year.

Cut the weeds and till the soil like you would normally do. Place a layer of newspaper about 5 sheets thick, around the plants. Cover the newspaper with lawn clippings or straw. The weeds can not get through the news print, so you will have very few weeds to contend with, mostly right next to the stem of the vegetable plant. Water passes through the newsprint and grass clippings or straw, and keep the soil moist.

By fall, the newspaper and most of the clippings / straw will be composted into the soil, making your garden spot better next year.

If you have trees, place the leaves that fall over your garden site. They keep the weeds from returning during the winter, and can be tilled under in the spring, which provides more rich compost for your soil..

You spray with a glyphosate based weedkiller. This gets to the roots and works well. I does not poison the soil either.

It is very easily and you just need to dig out and overturn the grasses' roots with the plough;then, spraying weedkiller pesticide and allow the land lie fallow for 2 weeks.
Yip

It sounds like many of the weeds are perennial (thick rooted). They will keep growing until you either dig out every last trace of root, or smother them with brown cardboard, black plastic or a really thick (>1 foot) layer of mulch. Smothering takes about 2 months. The best time to smother weeds with cardboard is actually late winter/early spring, as cardboard tends to rot over winter. Thick black plastic won't rot and it will prevent rain washing nutrients out, so this is another strategy, if you want to spend some money on the problem. Clear plastic can cook (solarise) the weeds with sunlight IF you are in a hot sunny climate. If it's a cool climate clear plastic will do very little. Rotovaters will chop up the perennial weed roots into tiny pieces which will all grow into new plants, so by itself it's not a good strategy. Cutting weeds to ground level will encourage grass to dominate, which is often easier to eradicate.

My suggestion is to dig a bed really thoroughly, take out any roots, (you could even dig a pit, pile all the earth on a tarp and then sieve it back in through a 1/2 inch mesh soil sieve, to take out any weed roots and stones) transplant the asparagus into there (making sure you don't move any weed roots with it and mulch thoroughly with cardboard etc. You may well leave some bits of asparagus behind, but don't worry, it grows well from root cuttings. This could be a good opportunity to multiply it. Give the new plants a year or so to recover before you seriously harvest.

Other things you may have planted in amongst weeds can also be transplanted into a carefully dug bed to free them from weeds, but use your judgement. Transplanting is a big shock, especially in hot dry weather, or if the seedlings are too small. Sometimes it's better to leave them in place and just battle the weeds hard.

It may not be too late to buy in pumpkin, cucumber, tomato plants etc. and if you really thoroughly dig out the weeds, you could plant them now. There are also leafy vegetables which you can plant for autumn and winter.

Dig as many other beds as your time allows. You could also smother/solarise some areas for autumn planting.

If you aren't in a desperate hurry, strim/scythe/mow the rest of the garden and pile up the weed cuttings in one place about 18 inches thick. After a month or two turn the heap and move it to another place. The ground underneath should be clear. Try to turn the cut areas into something resembling lawn (which will slowly reduce the amount of really nasty persistent weeds). In late winter you can prepare more ground for vegetables by smothering with cardboard, and then dig during the spring. Otherwise you can raise seedlings in pots or on a clean seed bed and transplant them through holes in the cardboard (some veggies like this, others like carrots, corn and beans hate it).

Remember also that weeds do a lot of good for the soil, they are often deep rooting, bringing up minerals from the subsoil, and provide ground cover. They are nature's response to bare, eroding soil. Half heartedly pulling up weeds and leaving the soil semi bare or totally bare over winter will do more harm than good, the soil will suffer and the worst weeds will flourish. If you decide to dig everything up but won't be seriously planting until next year, consider planting a green manure crop to protect the soil and then dig or rotovate in spring. Perennial weeds should be eradicated, but best to work with nature not against her, and understand WHY these weeds are there. Don't make your vegetable garden bigger than you can handle. The more the soil is bare and exposed to rain, the more manure and compost you will have to put back in.

You haven't said where you are and what kind of soil you have (clay, loam, sand etc.). This makes a difference. Clay soils shouldn't be dug when wet, and this influences your weed control strategy for spring.

Probably the best way is to rototill it out and place black plastic or black carpet(that stuff in the gardening store I can't think the name of).

Mow the weeds down to the ground with a string trimmer. Rake it clean, use the weeds to start a compost heap. Cultivate the soil with a tiller. Get a soil test, add in what they recommend, and a truckload of aged manure, if you can get it. Till it again, and then plant. After the plants get a few inches high, keep the weeds down with a few inches of natural mulch like shredded leaves, straw, or hay.