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Pack the eggs one dozen at a time.
Literal
Egg [object-を] one-dozen-each pack.
~ずつ ('each / at a time / per unit') distributes a quantity across repeated actions or recipients — 'X at a time.' Note the archaic spelling づつ instead of modern ずつ — still accepted but older. ダース is a katakana loanword from English 'dozen,' standard Japanese unit for eggs by the box.