Preheat your oven to 350°F (175°C). Place the 1/2 cup of unsalted butter in a 9x13 inch baking dish and put it in the oven to melt while the oven preheats, which should take about 5-7 minutes. Remove the pan from the oven once the butter is fully melted and golden.
In a medium mixing bowl, whisk together 1 cup of all-purpose flour, 2 teaspoons of baking powder, 1/4 teaspoon of salt, and 1 cup of granulated sugar until they are fully combined and no lumps remain.
Pour 1 cup of whole milk and 1 teaspoon of vanilla extract into the dry ingredients. Stir gently with a wooden spoon or spatula until a smooth, pourable batter forms, being careful not to overmix.
Arrange the sliced bananas evenly over the melted butter in the bottom of the baking dish. Pour the prepared batter over the bananas, spreading it gently with the back of a spoon to cover them completely.
Sprinkle 1 cup of packed brown sugar evenly over the top of the batter. Carefully pour 1 cup of boiling water over the entire surface of the cobbler. Do not stir.
Bake for 40 to 45 minutes, or until the top is golden brown, the edges are bubbling, and a cake tester inserted into the center comes out clean. The cobbler will have a cake-like top and a saucy bottom layer.
Notes
Do not stir after adding boiling water. Serve warm with vanilla ice cream.