1. Slice top 1/2 inch off 12 clementines; reserve. Cut around flesh and scoop out into a sieve set over a bowl; reserve skins, being careful not to tear them. Press flesh to extract juice. Squeeze in juice from tops. (You'll have about 1 cup.) Halve and juice remaining clementines (to yield 2 cups).

2. Bring 1/4 cup sugar, 1/4 cup water, and ginger to a boil in a pan, stirring, until sugar is dissolved. Remove from heat; let stand 30 minutes. Discard ginger. Stir in clementine and lemon juices; transfer to a nonreactive 8-inch square baking dish. Freeze until solid, about 3 hours.

3. Meanwhile, brush clementine skins with water. Roll in remaining 1/4 cup sugar to coat. Freeze until solid, about 2 hours.

4. Scrape granita with a fork to fluff. Spoon granita into clementine "cups." Freeze 10 minutes before serving.