Preheat your oven to 375°F (190°C).

In a medium saucepan, melt 1/2 cup of unsalted butter over medium heat. Stir occasionally until the butter turns golden brown and emits a nutty aroma. Remove from heat immediately.

In a large bowl, combine the sliced peaches, 1/2 cup granulated sugar, and 1 teaspoon vanilla extract. Toss to coat the peaches evenly.

Pour the browned butter over the peach mixture, gently mixing to combine, then pour this fragrant blend into a 9x13 inch baking dish.

In another bowl, whisk together 1 cup all-purpose flour, 1/2 cup granulated sugar, 1 teaspoon baking powder, and 1/2 teaspoon salt.

Add 1/2 cup milk and 1/4 cup melted butter to the dry ingredients. Stir until just combined; be cautious not to overmix.

Spoon the batter over the peach mixture, spreading it evenly across the surface.

Bake in the preheated oven for 35-45 minutes, or until the topping is golden brown and the peaches are bubbly.

Let it cool slightly before serving.