Preheat the oven to 350°F (176°C) and butter and flour a 9"x5" loaf pan. In a bowl, mash the bananas with a masher and set aside.
In a large bowl, whisk together 1 cup of flour, baking soda, and salt.
In another large bowl, combine 3/4 cup sugar, melted butter, buttermilk, vanilla extract, 1 egg, and 1 egg yolk. Mix until smooth. Pour the wet ingredients over the dry ingredients and stir until combined, then fold in the mashed bananas.
To make the cream cheese filling, combine the remaining egg, cream cheese, 2 tablespoons of flour, and 2 tablespoons of sugar in a small bowl. Stir until well blended and smooth.
Transfer half of the banana bread batter to the prepared loaf pan. Then, dollop the cream cheese filling on top and smooth it into an even layer using a spatula. Finally, pour the remaining batter over the cream cheese filling, spreading it evenly to cover.
Bake the banana bread until it's deeply golden and a toothpick inserted in the center comes out clean, about 1 hour. Once done, let it cool in the pan for at least 10 minutes before slicing and serving.