Preheat the oven to 350°F. Grease a 9x13-inch baking dish with butter or nonstick spray.
Brown the butter: Melt the butter in a saucepan over medium heat. Continue cooking, stirring often, until it foams and the milk solids turn golden brown and smell nutty, about 5–7 minutes. Remove from the heat and let cool for a few minutes.
Prepare the filling: In a large bowl, combine the sliced peaches, brown sugar, granulated sugar, cornstarch, lemon juice, vanilla, cinnamon, nutmeg, and a pinch of salt. Toss until evenly coated, then pour into the prepared baking dish.
Make the topping: In another bowl, stir together the oats, flour, brown sugar, cinnamon, and salt. Pour the browned butter over the mixture and stir until large crumbles form.
Sprinkle the topping evenly over the peaches, covering the entire dish.
Bake for 40–50 minutes, or until the topping is deep golden brown and the peach filling is bubbling around the edges.
Allow the crisp to rest for 15–20 minutes before serving so the filling can thicken.