Preheat the oven to 350 degrees F (175 degrees C). Grease a 9x5-inch loaf pan with cooking spray.
Make bread: Combine melted butter, brown sugar, egg, sour cream, white sugar, and vanilla in a large bowl; whisk until smooth. Mash in bananas using a fork. Whisk in flour, baking soda, baking powder, and salt until just combined.
Make filling: Whisk together cream cheese, egg, white sugar, and flour in a medium bowl until smooth.
Pour 1/2 of the banana batter into the prepared loaf pan. Spread with cream cheese filling. Pour remaining banana batter on top.
Bake in the preheated oven until top is firm and bounces back when gently pressed, about 50 minutes. Cool in the pan for about 5 minutes.