Place the butter in a light-colored saucepan over medium heat. Swirl the pan occasionally as the butter melts and begins to foam. Continue cooking for about 4 to 5 minutes, watching closely, until the butter turns a deep golden amber color and smells nutty.
Immediately pour the brown butter into a large heatproof bowl and let it cool for about 15 minutes until warm but not hot.
While the butter cools, whisk together the gluten free flour blend, baking soda, and fine sea salt in a medium bowl. Set aside.
Add the dark brown sugar and granulated sugar to the warm brown butter. Whisk vigorously until smooth and slightly thickened, about 1 minute. Whisk in the eggs one at a time, then add the vanilla extract and mix until the dough looks glossy and smooth.
Add the flour mixture to the wet ingredients and fold with a rubber spatula until just combined with no dry streaks remaining. Fold in the chocolate chunks. Cover the bowl tightly with plastic wrap and refrigerate for at least 1 hour or up to 48 hours.
Preheat the oven to 375 degrees F and line two baking sheets with parchment paper. Scoop the dough into generous 2-tablespoon balls and place them about 3 inches apart on the prepared sheets.
Bake for 10 to 12 minutes until the edges are set and golden brown but the centers still look slightly underdone and puffy. Immediately sprinkle each cookie with a small pinch of flaky sea salt.
Let the cookies cool on the baking sheet for 10 minutes. The centers will sink slightly and become perfectly chewy as they cool. Transfer to a wire rack to cool completely.