Step 1: Preheat your oven to 375°F (190°C). Lightly grease an 8x8 or 9x9 inch baking dish.
Step 2: In a large bowl, gently toss the sliced fresh peaches with 1/2 cup light brown sugar, cornstarch, and lemon juice until the peaches are well coated. Set aside.
Step 3: In a small saucepan, melt the 1 cup of unsalted butter over medium heat. Continue to cook, stirring occasionally, until the butter turns a rich amber color and small brown bits appear at the bottom (this is the brown butter). Immediately remove from heat and pour into a heatproof bowl to cool slightly.
Step 4: In a separate medium bowl, whisk together the all-purpose flour, old-fashioned rolled oats, granulated sugar, 1/4 cup light brown sugar, ground cinnamon, ground nutmeg, and salt.
Step 5: Pour the slightly cooled brown butter into the dry ingredients. Use your fingers or a pastry blender to mix until coarse, clumpy crumbs form. Do not overmix; you want varying sizes of crumbles.
Step 6: Pour the prepared peach mixture evenly into the greased baking dish.
Step 7: Sprinkle the brown butter oat crumble topping generously over the peaches, covering them completely.
Step 8: Bake for 35-45 minutes, or until the topping is golden brown and the peach filling is bubbly and tender when pierced with a knife. If the topping browns too quickly, you can loosely tent it with foil.
Step 9: Remove from oven and let cool for at least 15-20 minutes before serving. This allows the juices to thicken. Serve warm, perhaps with a scoop of vanilla ice cream.