Preheat your oven to 350°F. Line an 8x8-inch baking pan with parchment paper, leaving some overhang for easy removal.
In a saucepan over medium heat, melt the butter, stirring continuously until it turns golden brown and smells nutty. Remove from heat and let cool slightly.
In a large bowl, whisk together the brown butter, brown sugar, and granulated sugar until smooth. Add the egg, egg yolk, and vanilla extract, and mix until fully combined.
In a separate bowl, whisk together the flour, oats, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, stirring until just combined. Fold in the chocolate chips.
Transfer the dough into the prepared baking pan, spreading it evenly. Sprinkle with flaky sea salt if desired. Bake for 25-30 minutes, or until the edges are golden brown and a toothpick inserted in the center comes out with a few moist crumbs.
Let the bars cool in the pan for at least 20 minutes before slicing. Enjoy warm or at room temperature!
