Brown the Butter: Melt 10 tablespoons of unsalted butter in a non-stick pan over medium-low heat. Stir until it turns brown and smells nutty.
Add Sugars: Lower the heat to its lowest setting. Add ½ to ⅔ cup packed brown sugar and ¼ cup white sugar. Whisk for 1-2 minutes, then remove from heat.
Mix Wet Ingredients: Let the mixture cool for 5 minutes. Add 1 large egg, 1 egg yolk, 1 teaspoon vanilla bean paste, and 1 teaspoon instant coffee dissolved in 1 teaspoon hot water. Whisk until smooth and creamy.
Prepare Dry Ingredients: In a separate bowl, sift together 1⅓ cups all-purpose flour, ¾ teaspoon salt, ½ teaspoon baking powder, and ¼ teaspoon baking soda. Mix in 4 ounces each of chopped dark and milk chocolate.
Combine: Fold the dry ingredients into the wet mixture in two batches. Mix until just combined, being careful not to overmix.
Scoop and Chill: Use a 2-tablespoon cookie scoop to divide the dough into 10 balls. Add extra chocolate chunks on top. Refrigerate for at least 2 hours, preferably overnight.
Bake: Preheat oven to 375°F (190°C). Place chilled cookie balls on a parchment-lined baking sheet, leaving 3 inches between each. Bake for 11-13 minutes.
Finish: Sprinkle baked cookies with flaky sea salt if desired. Let cool on the baking sheet for 10-15 minutes before serving.