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.