Preheat the oven to 400 degrees.
In a 9-inch pie pan, stir together the fruit, brown sugar, 2 tablespoons granulated sugar, the cornstarch, and cinnamon.
In a bowl, stir together the flour, 2 1/2 tablespoons granulated sugar, the baking powder, baking soda, and salt. With a fork, blend in the butter until the mixture resembles coarse crumbs.
In a small bowl, whisk together the yogurt, milk, and vanilla and stir into the flour mixture until just combined. Knead the dough lightly on a work surface and pat lightly until it is about 3/4 inch thick.
Drop the dough in 4 mounds over the fruit, sprinkle the remaining 1/2 teaspoon sugar over the dough, and bake in the middle of the oven until the biscuits are golden and cooked through, about 20 to 25 minutes.