Polenta Cake with Blackberries and Fresh Corn

Name: Polenta Cake with Blackberries and Fresh Corn
Ingredients: 1 cup plus 2 Tbs. unbleached all-purpose flour, plus more for pan, divided; 2/3 cup stone-ground yellow cornmeal or polenta; 2/3 cup sugar; 2 tsp. baking powder; 1/2 tsp. coarse sea salt; 6 oz. firm silken tofu, drained (about 2/3 cup); 1/3 cup olive oil; 1/2 cup almond milk; 3/4 cup corn kernels from freshly shucked corn on the cob; 1 cup blackberries (about 4 oz.); 1 tsp. confectioners’ sugar; 3 cups blackberries (about 12 oz.), divided; 1/4 cup sugar; 1 Tbs. lime juice; 2 tsp. chopped fresh thyme, plus sprigs for garnish; 11/2 tsp. grated lime zest