Preheat the oven to 350F; line an 8 by 8-inch pan with 2 pieces of parchment paper.
Add the butter to a small skillet over medium heat. Cook until it smells nutty and brown bits form in the bottom; this takes approximately another 5 minutes after the butter is melted. Cool the butter to room temperature (very important so you don’t melt the chocolate chips).
Stir together the cooled brown butter (including the brown bits), brown sugar, egg, vanilla, flour, salt, and chocolate chips.
Pour the batter into the prepared pan and smooth out the top. Sprinkle the flaky sea salt on top.
Bake until golden, about 20 to 25 minutes, being careful not to over-bake.
Cool completely before cutting.