Preheat oven to 350 degrees F (175 degrees C). Grease a 9x5-inch loaf pan with cooking spray.
Combine butter, brown sugar, 1 egg, sour cream, 1/4 cup white sugar, and vanilla extract in a large bowl; whisk until smooth. Mash in bananas using a fork. Whisk in 1 cup flour, baking soda, baking powder, and salt until just combined.
Whisk cream cheese, 1 egg, 1/4 cup white sugar, and 3 tablespoons flour together in a separate bowl.
Pour half of the banana mixture into the prepared loaf pan. Spread cream cheese mixture evenly on top. Pour remaining banana mixture over cream cheese mixture.
Bake in the preheated oven until top is firm, about 50 minutes. Cool in the pan, about 5 minutes.