Preheat your oven to 350°F (175°C). Grease a 9x5 inch loaf pan.
In a mixing bowl, whisk together the flour, baking soda, and salt.
In another bowl, cream the butter and brown sugar until light and fluffy. Add the eggs one at a time, beating well after each addition. Stir in the mashed bananas and vanilla extract.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
In a separate bowl, beat the cream cheese, granulated sugar, and egg until smooth.
Pour half of the banana bread batter into the prepared loaf pan. Spread the cream cheese mixture evenly over the batter. Top with the remaining banana bread batter.
Bake in the preheated oven for 60 minutes, or until a toothpick inserted into the center comes out clean. Let cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.