Prepare. Preheat the oven to 375°F (190°C) with the rack in the center. Then, line a spare baking sheet with foil to catch any drips from the cobbler during baking.
Precook the filling. Combine the sliced peaches, granulated sugar, brown sugar, cinnamon, cornstarch, and lemon juice in a medium bowl. Next, transfer the mixture to an 8 or 9-inch square or round baking dish and cover tightly with foil. Bake the filling for 20 minutes to allow it to thicken.
Make the topping. While the filling is baking, prepare the topping. In a medium bowl, cream the softened butter and sugar using a medium-tailer until light and fluffy. Beat in the egg and vanilla extract until well combined. Gradually add the flour, baking powder, and a pinch of salt, mixing on low speed until just incorporated.
Assemble. Once the filling is done, remove it from the oven and discard the foil. Drop the sugar cookie topping by 2-tablespoon-sized dollops over the peach filling, spacing them evenly.
Bake and serve. Return the cobbler to the oven and bake, uncovered, for 30-40 minutes, or until the topping is golden brown and the filling is bubbling. During the last few minutes of baking, sprinkle coarse sugar on top if desired. Allow the cobbler to cool at room temperature for a few minutes before serving.