01 - Preheat the oven to 350°F (175°C).
02 - In a large mixing bowl, gently toss the banana slices with granulated sugar, brown sugar, 2 tablespoons melted butter, lemon juice, cinnamon, nutmeg, salt, and vanilla extract until evenly coated.
03 - Spread the banana mixture evenly into a greased 8-inch square baking dish.
04 - In a separate bowl, whisk together the flour, 1/2 cup granulated sugar, baking powder, and 1/4 teaspoon salt. Stir in the milk and 1/3 cup melted butter until just combined; do not over-mix.
05 - Spoon the cobbler batter evenly over the banana mixture. Gently spread to cover most of the filling.
06 - Bake for 35 to 40 minutes, or until the topping is golden brown and a toothpick inserted into the crust comes out clean.
07 - Let the cobbler cool slightly before serving warm. Optionally, top with vanilla ice cream or whipped cream.