feel 後面可以直接接動詞嗎?

"that (you felt) went beyond the call of duty" is a relative clause referring to the OBJECT of the verb "have done" in the main clause.

As the sentence is a question using there is (i.e delayed subject), the OBJECT would be the pronoun "anything".
---> Is there anything (that) subject + verb + (that) + relative clause ?
= Have "subject" ever done anything that (you felt) went beyond the call of duty?

"you felt" (also "you thought", "you believed", ...) is not actually part of the relative clause but an insertion that can be omitted, or placed somewhere else in the sentence.
(for example, Is there anything, you felt, any of ....?)

Now, we can put the question into an affirmative statement (indicative) to better understand the structure, as:
- Some of your English teachers have indeed done something that went beyond the call of duty to help you with your English.
- None of your English teachers has ever done anything that went beyond the call of duty to help you with your English.

Is there anything any of your English teachers have ever done to help you with English that you felt went beyond the call of duty. In the main sentence "Is there anything any of your English teachers have ever done to help you with English went beyond the call of duty." , while the" that you felt" is only a supplementary clause to modify the main sentence. Yip

因為(錯誤/以為可以)省略了"英語老師(們)

Is there anything any of your English teachers have ever done to help you with your English that you felt "they have" went beyond the call of duty?

以為go beyond the call of duty是一組慣用的phase便當作了是名詞或片語。