Step 1: Brown the Butter. In a saucepan over medium heat, melt the unsalted butter, stirring frequently, until it turns golden brown and develops a nutty aroma. This should take about 5-7 minutes. Once done, remove it from heat and let it cool slightly.
Step 2: Mix the Dry Ingredients. In a large mixing bowl, whisk together the cake flour, bread flour, baking soda, baking powder, and kosher salt.
Step 3: Combine Sugars and Butter. In another bowl, combine the browned butter with the light brown sugar and granulated sugar. Blend until smooth and creamy.
Step 4: Add Eggs and Vanilla. Add the eggs and egg yolk, one at a time, mixing well after each. Stir in the vanilla extract.
Step 5: Combine Wet and Dry Ingredients. Gradually add the dry mixture to the wet ingredients, mixing until just combined.
Step 6: Fold in the Chocolate and Toffee. Gently fold in the dark chocolate chips and toffee bits using a spatula.
Step 7: Chill the Dough. Cover the dough and chill in the refrigerator for at least 30 minutes.
Step 8: Preheat the Oven. While the dough chills, preheat your oven to 350°F (175°C).
Step 9: Scoop and Bake. Portion the dough onto the baking sheets, leaving space between each cookie. Sprinkle with flaky sea salt.
Step 10: Bake Until Golden. Bake for 10-12 minutes, or until edges are golden brown but centers are soft. Cool on baking sheets before transferring to a cooling rack.