Preheat the oven to 350°F (175°C). Grease a 9-by-13-inch (23-by-33-cm) baking pan or line it with parchment paper.

In a small bowl, combine the flour, brown sugar, cinnamon, and salt. Cut in the cold butter using a pastry cutter or your fingertips until coarse crumbs form. Refrigerate while preparing the batter.

Beat the softened cream cheese and sugar until smooth. Mix in the egg and vanilla until fully combined. Set aside.

In a medium bowl, whisk together the flour, baking powder, baking soda, and salt.

In a large bowl, beat the softened butter and granulated sugar for 2 to 3 minutes until light and fluffy.

Beat in the eggs one at a time, followed by the vanilla. Mix in the mashed bananas until combined. Fold in the sour cream.

Add the dry ingredients in two additions, mixing only until no dry streaks remain. The batter should be thick and smooth.

Spread about two-thirds of the batter evenly into the prepared pan. Spoon the cream cheese filling over the batter and gently spread it into an even layer, leaving about a 1/2-inch border around the edges. Carefully dollop and spread the remaining batter over the filling.

Sprinkle the chilled streusel evenly across the top of the batter.

Bake for 45 to 50 minutes, or until the topping is golden brown and a toothpick inserted into the cake portion comes out clean or with a few moist crumbs. Avoid inserting the toothpick directly into the cream cheese layer when checking doneness.

Place the pan on a wire rack and cool for 20 minutes before slicing. The cream cheese filling will continue to set during this time.

Slice into 12 portions and serve slightly warm or at room temperature.