01 - Preheat oven to 350°F. Grease an 8-inch square baking dish with butter or nonstick cooking spray and set aside.
02 - Melt 2 tablespoons of unsalted butter in a large skillet over medium heat. Add the sliced bananas, granulated sugar, brown sugar, cinnamon, nutmeg, vanilla extract, lemon juice, and a pinch of salt. Cook for 3 to 5 minutes, stirring gently, until the bananas soften and release their juices into a thick, syrupy sauce. Remove from heat and spread the mixture evenly across the bottom of the prepared baking dish.
03 - In a medium mixing bowl, whisk together the flour, granulated sugar, baking powder, and salt. Pour in the whole milk, melted butter, and vanilla extract, then stir gently until just combined. Be careful not to overmix, as a slightly lumpy batter produces the best texture.
04 - Drop spoonfuls of the batter over the banana filling, spacing them evenly. Use a spatula to gently spread the batter, leaving some gaps exposed so the filling bubbles through during baking for a rustic, golden finish.
05 - Place the baking dish on the center rack and bake for 35 to 40 minutes, or until the topping is deeply golden brown and a toothpick inserted into the crust comes out clean.
06 - Allow the cobbler to rest for 10 to 15 minutes before serving. Scoop into bowls while still warm and top with vanilla ice cream or a dollop of whipped cream if desired.