Melt the unsalted butter in a light-colored saucepan over medium heat. Continue cooking, swirling the pan often, until the butter foams, turns golden, and releases a nutty aroma with brown specks at the bottom (5-7 minutes). Pour immediately into a large heat-proof bowl, including all the brown bits. Let cool for 25-30 minutes until no longer hot but still liquid.
While the butter cools, whisk together the all-purpose flour, baking soda, and fine sea salt in a medium bowl. Set aside.
Add the granulated and dark brown sugars to the cooled brown butter. Using a hand mixer or stand mixer, beat on medium speed for 1-2 minutes until thick and glossy. Beat in the eggs one at a time, then the vanilla extract, until fully combined.
With the mixer on low, gradually add the flour mixture, mixing just until a few dry streaks remain. Use a spatula to fold in the chocolate chips by hand until evenly distributed.
Cover the bowl and chill the dough in the refrigerator for at least 1 hour (up to 48 hours for deeper flavor).
Preheat your oven to 375°F (190°C). Line baking sheets with parchment paper. Use a 1.5-tablespoon cookie scoop to portion dough. Roll each into a smooth ball.
Roll a dough ball into a short, fat log about 2 inches (5 cm) long. Place it on the sheet. Use your thumb or a chopstick to press a deep indentation into the top center. Gently pinch the bottom of the log to form a point. You now have a heart shape. Repeat, spacing hearts 3 inches (7.5 cm) apart.
Bake one sheet at a time for 8-10 minutes, until edges are golden but centers are soft and puffy. If desired, press a few extra chocolate chips on top and sprinkle with flaky salt as soon as they come out of the oven.
Let cookies cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.