Preheat the oven to 350°F (175°C). Grease a 9×13-inch metal baking pan with nonstick spray or line it with parchment paper, leaving some overhang for easy removal. Set the pan aside.
Place the 1 cup unsalted butter (cut into tablespoon pieces) in a medium saucepan over medium heat. Stir often as the butter melts and foams, continuing to cook and stir until it smells nutty and turns brown with browned bits on the bottom (about 5 minutes).
Immediately scrape the browned butter into a large mixing bowl and let it cool for 5 minutes.
Add the 1 1/4 cups packed dark brown sugar and 1/4 cup granulated sugar to the cooled browned butter. Stir with a wooden spoon or spatula until the sugars are fully incorporated and the mixture is smooth.
Add the 2 large eggs and 1 tablespoon pure vanilla extract to the butter-sugar mixture. Stir until combined.
Add 2 cups all-purpose flour, 3/4 teaspoon kosher salt, 1/2 teaspoon baking soda, and 1/2 teaspoon baking powder to the bowl. Stir just until the dry ingredients are incorporated — do not overmix.
Fold in 1 cup chocolate chunks and 1 cup chocolate chips until evenly distributed.
Spread the dough evenly in the prepared 9×13-inch pan using a spatula, smoothing the top. Bake for 18 to 24 minutes, or until the bars are set and golden brown at the edges (the center may still be slightly soft — they will continue to set as they cool). Avoid overbaking.
Remove the pan from the oven, immediately sprinkle the top with flaky sea salt, and transfer the pan to a wire rack. Let the bars cool in the pan until set and easy to slice (about 30–45 minutes) before cutting into squares.