Step 1: Preheat your oven to 375°F (190°C). Lightly grease a 9-inch pie dish or an 8×8 inch baking dish.
Step 2: In a large bowl, combine the sliced nectarines, granulated sugar, cornstarch, lemon juice, and a pinch of salt. Toss gently until the nectarines are evenly coated. Pour the mixture into your prepared baking dish, spreading it evenly.
Step 3: In a small saucepan, melt the unsalted butter over medium heat. Continue cooking, swirling occasionally, until the butter turns a rich golden-brown and smells nutty (this usually takes 5-7 minutes). Be careful not to burn it. Remove from heat immediately.
Step 4: In a separate medium bowl, combine the all-purpose flour, brown sugar, old-fashioned rolled oats, ground cinnamon, and 1/4 teaspoon salt. Whisk to combine thoroughly.
Step 5: Pour the warm brown butter over the dry ingredients in the bowl. Use a fork or your fingertips to mix until the mixture is crumbly and well combined. Some larger clumps are desirable.
Step 6: Evenly sprinkle the brown butter oat crumble topping over the nectarine filling in the baking dish.
Step 7: Place the dish on a baking sheet (to catch any potential drips) and bake in the preheated oven for 35-45 minutes, or until the topping is golden brown and the filling is bubbly and tender when poked with a knife.
Step 8: Remove from the oven and let the crumble cool for at least 15-20 minutes before serving. This allows the filling to set. Serve warm, perhaps with a scoop of vanilla ice cream or whipped cream.