Heat oven to 350°F. Add 4 tbsp melted butter to a 9x13-inch baking dish and warm until sizzling.
Toss apples with lemon juice, 1/2 cup sugar, brown sugar, cinnamon, nutmeg, and cornstarch; set aside to juice.
Whisk flour, 3/4 cup sugar, baking powder, and salt in a bowl.
Stir in milk and vanilla until a smooth batter forms.
Pour batter into the hot, buttered dish without stirring it into the butter.
Scatter apple mixture evenly over the batter and dot with the 2 tbsp cold butter; do not stir.
Bake 40–45 minutes, until the topping is golden and juices bubble. Rest 10 minutes.
Serve warm. Add vanilla ice cream or a dollop of whipped cream if you like.