He usually moves the eggs quickly but very carefully which adverb modifies Another adverb

Very

Very is the adverb that is modifying another adverb. In this sentence, the adverbs are usually, quickly, very, and carefully. Usually modifies the verb moves telling how often he moves the eggs. Quickly and carefully both modify the verb moves as well. They describe how he moves - quickly and carefully. Very is an adverb that clarifies how carefully the speaker is moving.

Very is modifying carefully