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In the blink of an eye, the tennis-ball-sized dumplings completely vanished.
Literal
Blink [in the span of-うちに] tennis-ball-size [of-の] dumplings [subject-が] disappeared-completely.
瞬くうちに literally 'while blinking' is an idiomatic phrase for 'in the blink of an eye,' using the pattern ~うちに 'while/during.' The suffix ~大 attaches to a noun to mean 'the size of,' giving テニスボール大 'tennis-ball-sized' — a compact way to specify dimension without ほど or くらい. 消え去る is a compound verb combining 消える 'disappear' and 去る 'leave,' intensifying the disappearance to 'completely vanished.'