01 - Preheat the oven to 350°F. Grease an 8-inch square baking dish with butter or nonstick spray.
02 - In a large mixing bowl, gently toss the sliced bananas with granulated sugar, brown sugar, 2 tablespoons melted butter, lemon juice, cinnamon, nutmeg, salt, and vanilla extract until evenly coated. Spread the mixture in an even layer across the bottom of the prepared baking dish.
03 - In a separate bowl, whisk together the flour, sugar, baking powder, and salt. Pour in the milk and 1/3 cup melted butter, stirring gently until just combined. Be careful not to overmix; a few small lumps are fine.
04 - Spoon the batter evenly over the banana filling, spreading gently with a spatula to cover most of the surface. The batter will spread and bubble as it bakes.
05 - Bake for 35 to 40 minutes, or until the topping is golden brown and a toothpick inserted into the crust comes out clean. The banana filling should be bubbling around the edges.
06 - Allow the cobbler to cool for 10 to 15 minutes before serving. Serve warm on its own or accompanied by vanilla ice cream or freshly whipped cream.