Prepare the oven. Preheat the oven to 375°F with the rack in the center position. Then, line a spare baking sheet with aluminum foil to catch any drips during baking.
Prepare the berries. If you have fresh berries, rinse them gently with cold water and dry them with paper towels. If you're using frozen berries, you don't need to thaw them first. Mix the berries, sugar, and cornstarch in a bowl. Make sure the berries are coated evenly.
Precook the filling. Put the mixture in an 8x8-inch baking dish and cover it tightly with foil. Bake for 20 minutes to thicken the filling.
Make the topping. While the filling is baking, prepare the topping. In a large mixing bowl, use a hand mixer on medium speed to cream together the softened butter and sugar until the mixture is light and fluffy. Then, add the egg yolk and vanilla extract and beat until fully incorporated. Slowly add the flour and baking powder, mixing on low speed until just combined. Don't mix too much.
Assemble the cobbler. After the filling is done, take it out of the oven and throw away the foil. Then, use a tablespoon to spoon the sugar cookie topping over the fruit, making sure to space it evenly across the surface.
Bake and serve. Put the dish back in the oven and bake for 35-45 minutes, or until the top is golden brown and the filling is bubbly. If you like, add a little sugar on top during the last few minutes of baking. Take it out of the oven and let the cobbler sit at room temperature for a few minutes before serving. Serve it warm with a scoop of vanilla ice cream if you have some.Tip: If you notice the topping is getting too brown while the filling is still cooking, just cover the cobbler loosely with a piece of foil. This will prevent further browning while the filling finishes bubbling away.