Preheat the oven to 450°F.
Pour the melted butter into an 8x8-inch baking dish. Place the dish on a sheet pan to catch any butter that may bubble over during baking.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, and brown sugar.
In a small bowl, mash the bananas until mostly smooth. Stir in the vanilla and buttermilk.
Pour the banana mixture into the dry ingredients. Stir just until combined. The dough should be thick and scoopable, not thin like cake batter. If it seems too wet, add 1 to 2 tablespoons of flour. If it seems too stiff, add 1 tablespoon of buttermilk.
Spoon the dough over the melted butter and gently spread it into an even layer, reaching the edges of the pan. Do not stir the butter into the dough.
Using a knife, gently score the dough into 9 squares.
Bake for 18 minutes. If the edges are getting too dark, loosely tent the pan with aluminum foil, then continue baking for another 7 to 9 minutes, or until the biscuits are golden and set. The total baking time will be 25 to 27 minutes.
Let the biscuits rest in the pan for 10 minutes so the butter can finish soaking into the biscuits. Serve warm.