01 - Preheat the 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 the mixture becomes syrupy and saucy. Remove from heat and spread the filling 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 until just combined. Be careful not to overmix, as a slightly lumpy batter produces a more tender topping.
04 - Spoon the batter in generous dollops over the banana filling. Use a spatula to gently spread it outward, leaving some gaps exposed so the filling bubbles through during baking for a rustic, golden finish.
05 - Place the dish on the center rack and bake for 35 to 40 minutes, or until the topping is deep golden brown and a toothpick inserted into the crust comes out clean. The filling should be bubbling around the edges.
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.