Preheat oven to 350° with top rack in center and another rack beneath it. (If using a glass baking dish, set oven at 325° and set top rack in upper third of oven). Make filling: In a large bowl, combine granulated sugar, tapioca, cornstarch, and cardamom. Stir in 1/4 cup water, the orange zest and juice, lemon zest and juice, and vanilla seeds. Gently stir in berries. Let stand 15 to 20 minutes, stirring occasionally, for tapioca to begin to soften.
Meanwhile, make biscuits: In a bowl, stir together flour, granulated sugar, baking powder, salt, and soda. Add butter and cut in with a pastry blender until pieces are pea-size. Stir in sour cream and 1 tbsp. whipping cream until dough is evenly moistened but still shaggy. Gather dough into a ball and knead on a lightly floured work surface just until smooth, 4 to 6 turns.
Re-flour surface and pat or roll dough into a 6-in. square about 1/2 in. thick. Cut straight down with a long knife into 2-in. squares to make 9; or cut straight down with a 2-in. round biscuit cutter, re-rolling scraps as needed.
Transfer fruit and juices to a 2- to 2 1/2-qt. baking dish that’s 1 1/2 to 2 in. deep, such as an 8-in. square. Set biscuits slightly apart on fruit. Brush biscuits with remaining 2 tsp. cream and sprinkle with sparkling sugar.
Set cobbler on upper oven rack and set a rimmed baking sheet on rack beneath it to catch drips. Bake until filling begins to bubble in center and biscuits are well browned, about 55 minutes, 1 1/4 hours for a glass dish. Let cool on a rack at least 1 1/2 hours. Serve warm or at room temperature.