Preheat the oven to 350°F. Line a loaf pan with parchment paper leaving a few inches hanging over the edge for easy removal later.
In a large bowl, cream the butter and brown sugar using an electric mixer until light and fluffy, about 3 minutes.
Add the mashed banana, eggs, and vanilla. Mix until combined.
Add the flour, cinnamon, baking soda, and salt. Mix on low until just combined, being careful not to overmix.
Fold in 3/4 cup of the white chocolate chips.
Pour the batter into the prepared pan, spreading it out evenly. Scatter the top with the 2 tablespoons of white chocolate chips.
Bake for 60 to 65 minutes, or until a toothpick inserted in the center comes out clean. Let the banana bread cool in the pan for 15 minutes. Remove it from the pan and let it cool completely on a wire rack, approximately 2 hours.