Dairy

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Middle English deierie, from deie ‘dairymaid’ (in Old English dǣge ‘female servant’), of Germanic origin; related to Old Norse deigja, also to dough and to the second element of Old English hlǣfdige (see lady).

wiktionary

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Origin 1250–1300 (Middle English daierie and other forms), from dey(“dairymaid”) + -ery.

etymonline

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dairy (n.)

c. 1300, daerie, "building for making butter and cheese; dairy farm," formed with Anglo-French -erie (from Latin -arius; see -ery) affixed to Middle English daie (in daie maid "dairymaid"), which is from Old English dæge "kneader of bread, housekeeper, female servant" (see dey (n.1)). The pure native word was dey-house (mid-14c.). Meaning "branch of farming concerned with the production of milk, butter, and cheese" is from 1670s. Later also "shop where milk, butter, etc. are sold."