Preheat The Oven & Prep Your Pan. Start by preheating the oven to 325 degrees F. Line an 8x8 metal baking pan with parchment paper. (Stick with a metal pan, as a glass pan or ceramic pan will cause the brownies to bake longer and climb the sides.)
Melt Chocolate & Butter. Add the butter and 2/3 cup dark chocolate chips to a saucepan and melt on low heat until smooth. (Or, add them to a microwave-safe bowl and melt in 20-second increments until smooth.) Set the melted chocolate mixture aside to cool while you mix the eggs and sugar.
Dissolve The Sugar. Next, add the sugar, eggs, and egg yolk to a large mixing bowl. Whisk or beat the sugar and eggs 2-3 minutes with a hand mixer until the sugar is dissolved and the mixture is light yellow. (There should be VERY little grit when you rub the sugar/egg mixture between your fingers.)
Combine. Gradually mix the melted chocolate mixture into the eggs and sugar mixture until well blended. Mix in the vanilla.
Add Dry Ingredients. Then, add the flour, cocoa power, and salt to the brownie batter. Gently fold the dry ingredients in with a spatula until just a few floury patches remain. Set aside 1 Tablespoon of each kind of chocolate chips, then add the remaining dark chocolate chips, semisweet chocolate chips and white chocolate chips and gently fold in until the brownie batter is just combined.
Bake & Cool. Pour the brownie batter into the prepared pan. Sprinkle with the reserved chocolate chips (I like using 1 Tablespoon of each kind). Bake 25-30 minutes in preheated oven, or until the top is set and shiny, and a toothpick inserted in the center has some fudgy brownie on it. (If it comes out clean, the brownies are over-cooked. If it comes out with liquid brownie batter, the brownies aren't done yet.) Mine are usually done around 25-26 minutes. The brownies will continue to set as they cool. Set the brownies on a wire rack to cool completely. For the neatest cuts, chill the brownies in the fridge after they're cooled before cutting them into squares.
Serve & Store. Cut the brownies into squares or rectangles and enjoy right away. Store leftover brownies in an airtight container at room temperature 2-3 days, or in the freezer up to 2 months.