01 - Preheat oven to 350°F. Lightly 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 a uniform layer across the bottom of the prepared baking dish.
03 - In a separate bowl, whisk together the flour, granulated sugar, baking powder, and salt. Pour in the whole milk and 1/3 cup melted butter, stirring gently until just combined. Avoid overmixing to keep the topping tender.
04 - Spoon the batter in even dollops over the banana filling, then gently spread it with a spatula to cover most of the fruit surface.
05 - Bake on the center rack for 35 to 40 minutes, until the cobbler crust is golden brown and a toothpick inserted into the topping comes out clean.
06 - Allow the cobbler to rest for 10 minutes before serving. Serve warm on its own or topped with vanilla ice cream or freshly whipped cream.