Preheat oven to 350°F. Grease a 10-inch cast iron skillet, or another 2-quart baking dish (such as an 11 x 7-inch pan or a deep 8-inch or 9-inch square pan) with nonstick cooking spray.
Add peaches and blackberries to the prepared dish. Sprinkle with 1 cup of the granulated sugar, ¼ cup of the flour, and the almond extract. Toss gently to combine.
In a medium bowl, whisk together the remaining ½ cup of sugar, remaining 1 cup of flour, cornmeal, baking powder, and salt. Use a fork or pastry blender to cut in the butter until the mixture resembles coarse crumbs. Add the cream; stir gently just until the dough comes together.
Drop the dough by heaping tablespoonfuls evenly over the fruit mixture. Sprinkle with the sparkling sugar.
Bake until the topping is browned and the filling is hot and bubbly, about 45-55 minutes. Cover the dish loosely with foil as it’s baking if you notice that the top is browning too quickly. Let the cobbler cool for at least 10 minutes before serving.