1 1/2 cups all-purpose flour
1/2 cup yellow cornmeal (not stone-ground)
1 tablespoon sugar
1/4 teaspoon salt
1 stick (1/2 cup) cold unsalted butter, cut into pieces
1 1/2 tablespoons finely chopped fresh rosemary
4 to 5 tablespoons ice water
1/3 cup sour cream
1 cup mascarpone cheese (8 oz)
1/4 cup sugar
1 1/2 teaspoons finely grated fresh lemon zest
1/8 teaspoon salt
2 tablespoons red-currant jelly
1 tablespoon honey
1 1/2 lb fresh figs
Pulse together flour, cornmeal, sugar, and salt in a food processor. Add butter and rosemary and pulse until mixture resembles coarse meal with some small (roughly pea-size) butter lumps. Drizzle evenly with 4 tablespoons ice water and pulse until just incorporated.
Gently squeeze a small handful: If it doesn't hold together, add more water, 1/2 tablespoon at a time, pulsing after each addition and continuing to test.
Press dough evenly onto bottom and up sides of tart pan with floured fingers. Smooth dough with a small offset metal spatula or back of a spoon (floured if necessary), then roll a rolling pin over top of pan to trim dough flush with rim. Chill crust until firm, about 30 minutes.
Preheat oven to 400°F.
Bake crust in middle of oven until center and edges are golden, 25 to 30 minutes (don't worry if bottom of crust cracks), then cool in pan on a rack.
Prepare filling and assemble tart:
Whisk together sour cream, mascarpone, sugar, zest, and salt in a bowl.
Heat jelly and honey in a small saucepan over moderately low heat, whisking, until jelly is melted, about 4 minutes, then cool glaze slightly.
Remove side of tart pan and spread mascarpone cream in shell. Cut figs lengthwise into 1/4-inch-thick slices and arrange decoratively over cream. Brush figs with honey glaze.