01 - Preheat the oven to 350°F. Grease an 8-inch square baking dish with butter or nonstick cooking spray.
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 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.
04 - Spoon the batter in generous dollops over the banana filling. Use a spatula to gently spread it out, leaving some areas of the filling exposed for a rustic, golden finish.
05 - Bake for 35 to 40 minutes, or until the topping is golden brown and a toothpick inserted into the crust comes out clean.
06 - Let the cobbler cool for 10 to 15 minutes before serving. Serve warm with vanilla ice cream or whipped cream if desired.