Preheat your oven to 350˚F. Lightly grease a 2-quart baking dish with a small amount of butter to prevent sticking and set it aside while you prepare the rest of the cobbler.
Arrange the peeled, pitted, and sliced peaches evenly on the bottom of the prepared baking dish. Sprinkle 2 tablespoons of white sugar over the peaches. If your peaches are especially soft and juicy, you may want to toss them with a tablespoon of cornstarch before adding to the dish to help the filling thicken during baking.
In a mixing bowl, whisk together the all-purpose flour, 3/4 cup white sugar, baking powder, ground cinnamon, nutmeg, and a pinch of salt until fully combined. This ensures the leavening and spices are well distributed.
Melt the butter and then whisk in the vanilla extract. Pour this butter-vanilla mixture over the dry ingredients from Step 3. Stir until everything is incorporated and the batter is moist. The resulting mixture should resemble a thick cookie dough. I like to ensure every bit of flour is moistened–this helps create a perfect golden topping.
Using your hands, gently sprinkle the batter (from Step 4) evenly over the peach mixture (from Step 2) in the baking dish. Place the dish in the preheated oven and bake for 25 to 30 minutes, or until the topping is golden brown and the fruit filling is bubbling up around the edges.
Remove the cobbler from the oven and let it rest at room temperature for 15 minutes before serving. This cooling period allows the juices to thicken up. Serve warm, ideally with a scoop of your favorite ice cream for an extra treat!