Preheat the oven to 375 degrees. Line two rimmed baking sheets with parchment paper and set aside.
Place the butter in a large skillet (preferably not nonstick or cast iron so that you can more easily see the color change) and heat over medium heat, swirling the pan frequently. Once the butter has melted, continue to cook over medium low, swirling the pan frequently, until the butter has a nutty aroma and is golden brown in color. Transfer brown butter to a large mixing bowl and allow it to cool to room temperature.
Whisk rye flour, all-purpose flour, baking soda and fine salt together in a medium bowl. Set aside.
Add the brown sugar, granulated sugar to the cooled brown butter. Whisk thoroughly to combine. Add the vanilla, egg and egg yolk and whisk vigorously until the mixture is golden brown and smooth and homogeneous in texture .
Add the dry incredients to the wet ingredients. Stir until just combined. Add chocolate chips and stir until evenly distributed throughout the dough.
Drop two tablespoon-sized balls of dough onto each prepared baking sheet, leaving 2 inches between each cookie (about 8 cookies per sheet). Sprinkle each cookie with flaky sea salt. Bake for 12 minutes or until cookies are golden brown and puffed in the center. Transfer to a wire rack to cool. Repeat with remaining dough.