Pulse the flour, sugar, cornmeal, and salt in a food processor until combined. Add the butter and pulse until the mixture resembles coarse meal with a few pea-size clumps of butter remaining. Add the egg yolks, figs, and thyme and pulse just until the dough comes together.
Form the dough into two 12-by-1½-inch logs. Wrap tightly in plastic wrap and refrigerate until firm, about 2 hours and up to 2 days.
Heat oven to 350° F. Line 2 baking sheets with parchment.
Cut the logs into 3/8-inch slices. Place the cookies 1 inch apart on the baking sheets. Bake, rotating the sheets halfway through, 12 to 14 minutes. Let cool slightly on the baking sheets, then transfer to a wire rack to cool completely.