01 - Preheat the oven to 350°F. Grease an 8-inch square baking dish with butter or nonstick 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. 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. Stir gently until just combined, being careful not to overmix.
04 - Spoon the batter in generous dollops over the banana filling. Use a spatula to spread it gently, leaving some gaps exposed so the filling bubbles through during baking.
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 - Allow the cobbler to cool for 10 to 15 minutes. Serve warm with a scoop of vanilla ice cream or a dollop of fresh whipped cream.