Preheat the oven to 350°F (175°C). Grease and flour a 9-inch square baking pan.
In a medium bowl, whisk together 1 1/2 cups flour, baking powder, baking soda, salt, and 1 teaspoon cinnamon. Set aside.
In a large bowl, beat the softened butter, granulated sugar, and 1/4 cup brown sugar with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition. Mix in the vanilla extract.
Add the mashed bananas, sour cream, and milk to the butter mixture and stir until combined.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined. Do not overmix.
To make the streusel topping, combine 1/2 cup flour, 1/3 cup brown sugar, and 1 teaspoon cinnamon in a small bowl. Cut in the cold butter pieces with a pastry cutter or fork until the mixture resembles coarse crumbs.
Pour the batter into the prepared baking pan and spread evenly. Sprinkle the streusel topping evenly over the batter.
Bake for 45 to 50 minutes, or until a toothpick inserted into the center comes out clean.
Remove from the oven and let cool in the pan on a wire rack for at least 20 minutes before slicing and serving.