Melt butter in a medium, heavy saucepan over medium heat; cook, stirring occasionally, until the butter begins to turn golden brown and smells nutty, 6 to 8 minutes. Remove from heat and let cool, uncovered, for 10 minutes.

Transfer the butter to a large bowl. Add brown sugar and granulated sugar; beat with an electric mixer on medium speed until well combined, 2 to 3 minutes. Add eggs, 1 at a time, beating on low speed until just incorporated after each addition. Add vanilla; beat until just incorporated.

Sift all-purpose flour, whole-wheat flour, baking soda, baking powder and 1/2 teaspoon salt together in a medium bowl. Gradually add the flour mixture to the butter mixture, beating on low speed until incorporated, about 2 minutes, stopping to scrape down sides of bowl as needed. Using a spatula, stir in chopped chocolate until combined. Let the dough rest, uncovered, at room temperature for 20 minutes.

Meanwhile, preheat oven to 375°F. Line 3 baking sheets with parchment paper.

Using a 1 1/2-inch cookie scoop or 1 1/2 leveled tablespoons, scoop and drop the dough 2 inches apart on the prepared baking sheets. Sprinkle evenly with flaky sea salt.

Bake the cookies in 3 separate batches until golden, about 10 minutes per batch. Let cool on the baking sheet for 5 minutes. Transfer to wire racks and let cool completely, about 30 minutes.