In a bowl, with an electric mixer on high speed, beat egg whites until frothy. Beat in vanilla, sugar, and flour until well blended. Stir in coconut.
Drop dough in 1-tablespoon portions, about 2 inches apart, onto buttered and floured or cooking parchment- lined 12- by 15-inch baking sheets.
Bake in a 325° oven until macaroons are golden, about 20 minutes; if baking more than one pan at a time, switch pan positions halfway through baking. With a wide spatula, transfer macaroons to racks to cool completely.
In a heatproof bowl set over a pan of barely simmering water (bottom of bowl should not touch water), stir butter and chocolate often until smooth, 5 minutes. Hold a macaroon on one edge and dip other side into chocolate to coat half the cookie. Shake off excess chocolate. Set macaroon on a waxed paper-lined baking sheet. Repeat to dip remainder. Chill, uncovered, until chocolate is firm, about 30 minutes.
Note: Nutritional analysis is per cookie.
To store, chill cookies airtight for up to 1 week.