Step 1: Preheat your oven to 375°F (190°C). Lightly grease an 8x8 inch or 9-inch round baking dish.
Step 2: Prepare the peach filling: In a large bowl, combine the sliced peaches, granulated sugar, cornstarch, vanilla extract, cinnamon, and a pinch of salt. Toss gently until the peaches are evenly coated. Pour the mixture into the prepared baking dish.
Step 3: Prepare the brown butter: In a small, light-colored saucepan over medium heat, melt the unsalted butter. Continue cooking, swirling occasionally, until the butter turns a rich amber color and nutty aroma fills the air. Watch carefully to prevent burning. Remove from heat and let cool slightly.
Step 4: Prepare the oat topping: In a medium bowl, whisk together the flour, rolled oats, brown sugar, granulated sugar, cinnamon, and salt.
Step 5: Pour the slightly cooled brown butter over the dry topping ingredients. Use a fork or your fingertips to mix until the mixture is crumbly and resembles coarse meal with some larger clumps. Do not overmix.
Step 6: Evenly sprinkle the oat topping over the peach filling in the baking dish.
Step 7: Bake for 40-45 minutes, or until the topping is golden brown and the peach filling is bubbly and tender. If the topping starts to brown too quickly, you can loosely tent it with aluminum foil.
Step 8: Remove from the oven and let cool for at least 15-20 minutes before serving. This allows the filling to set slightly. Serve warm with a scoop of vanilla ice cream or whipped cream.