01 - Preheat oven to 350°F. Grease an 8-inch square baking dish with butter or nonstick cooking spray and set aside.
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 a single, even layer across the bottom of the prepared baking dish.
03 - In a separate medium 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. Take care not to overmix — a few small lumps in the batter are perfectly fine.
04 - Spoon the batter in even dollops over the banana filling, then use a spatula to gently spread it, covering most of the fruit beneath. Some gaps are welcome — the batter will expand as it bakes.
05 - Place the dish on the center oven rack and bake for 35 to 40 minutes, rotating once halfway through. The cobbler is done when the topping is deep golden brown and a toothpick inserted into the crust comes out clean.
06 - Remove from the oven and allow the cobbler to rest for 10 to 15 minutes before serving. Spoon into bowls while still warm. Pair with a scoop of vanilla ice cream or a dollop of freshly whipped cream, and finish with a drizzle of caramel sauce if desired.