Brown the Butter: Melt 1/2 cup unsalted butter in a light-colored saucepan over medium heat. Swirl occasionally for 4 to 5 minutes as it foams and sizzles. Once golden-amber with nutty toasted brown flecks, immediately pour into a heat-safe bowl and let cool for 10 minutes.
Toss the Apple Filling: Preheat oven to 350°F (175°C). In a large bowl, toss sliced apples with brown sugar, granulated sugar, lemon juice, arrowroot powder, cinnamon, nutmeg, and vanilla extract until glossy. Spread evenly into a 9-inch square baking dish or 10-inch cast-iron skillet.
Make Brown Butter Streusel: In another bowl, combine rolled oats, flour, brown sugar, chopped pecans, cinnamon, and sea salt. Pour the cooled nutty brown butter over the dry ingredients. Mix with a fork until large, crumbly clusters form.
Top & Bake: Scatter the streusel evenly over the apples, pressing a few clusters together for extra-crunchy texture. Bake at 350°F (175°C) for 38 to 42 minutes until the topping is deep golden-brown and the cinnamon apple juices bubble thickly around the edges.
Rest & Serve Warm: Allow the crisp to rest for 10 minutes so the juices settle. Scoop warm into bowls, top with a generous scoop of vanilla bean ice cream, and serve immediately.