Brown the butter: Place the butter in a medium saucepan over medium heat. Once the butter has melted, continue cooking, swirling the pan often, until the butter begins to foam. This will take about 6-8 minutes. Once the butter has turned a light brown color and smells nutty, remove from the heat and keep whisking until it turns a rich brown color. Transfer the butter to a bowl, making sure that you scrape down the brown bits from the bottom of the saucepan. Allow the brown butter to cool for about 30 minutes before using.
Place the brown butter in a large mixing bowl. Add the granulated and brown sugar and whisk to combine.
Next, add the egg yolk and vanilla extract to the butter/sugar mixture. Whisk well to combine.
Mash the banana with the lemon juice, and then add to the rest of the ingredients. Mix well to combine.
In a separate bowl, combine the flour, baking soda, and salt. Add the dry ingredients to the brown butter mixture and mix until combined.
Finally, stir in the white chocolate chips until they are evenly distributed throughout the cookie dough.
Use a cookie scoop (No. 40) to drop 1.5 tablespoon-sized dough balls onto a parchment lined baking sheet. Cover the cookie dough tightly with plastic wrap and chill for at least 1 hour. (If you are in a rush, you can chill the cookie dough for a minimum of 30 minutes).
Preheat oven to 350°F. Line two baking sheets with parchment paper.
Place the chilled cookie dough balls 2 inches apart (no more than 10 cookies per tray). Bake the cookies for 10-12 minutes, until they are golden and centers are still soft. Remove from oven and cool cookies on baking sheet for 5 minutes before transferring to a wire rack.