Preheat Oven to 350 degrees F. Line an 8x8 metal baking pan with parchment paper & set aside.
Melt Chocolate & Better. In a small saucepan on the stove, melt butter and 2/3 cup chocolate chips over low heat until smooth and glossy. Set aside to cool slightly.
Whip Sugar & Eggs. In a large bowl, combine sugar, brown sugar, eggs, and egg yolk. Use a hand mixer to beat the eggs and sugar together 2-3 minutes, or until the mixture is lighter in color and the sugar is fully dissolved. (You shouldn't feel any grit when you rub the mixture between your fingers.) Do NOT skip this step or your brownies won't have a shiny, glossy top. Add Chocolate Mixture. Slowly, mix the melted chocolate mixture into your whipped egg mixture, until smooth. Mix in the vanilla extract.
Add Dry Ingredients. Next, add gluten-free flour, cocoa powder, and salt. Gently stir until almost combined, with just a few dry spots left. Add the remaining 2/3 cup chocolate chips and gently fold in until well distributed.
Warm Dulce De Leche. In a small microwave-safe bowl, gently warm the dulce de leche 30-60 seconds, stirring every 15-20 seconds or so, until the dulce de leche has a smooth, pourable texture.
Assemble The Brownies. Set aside 1/2 cup brownie batter, then pour the rest of the batter into the prepared pan. Smooth into an even layer, paying special attention to the edges, making sure the batter has a nice seal around the edges of the pan. (You can even make a slight border by smoothing some of the batter up the sides slightly, if desired.) Pour dulce de leche over the brownie batter and smooth into an even layer. Dollop the remaining 1/2 cup brownie batter over the dulce de leche.
Swirl. Use a thin knife to make a few gentle swirls lengthwise and widthwise through the top brownie batter dollops and the dulce de leche (try not to swirl all the way down to the bottom of the pan, so you get a nice layer of dulce de leche in the center)
Bake Brownies at 350 degrees for 28-30 minutes, or until the tops look glossy and the center is just set.
Cool Completely before slicing. Transfer the brownies (using your parchment paper "handles") to transfer the brownies to a cutting board before cutting into slices or squares.
Serve & Store. Enjoy right away (or with a scoop of ice cream!) and store leftover brownies in an airtight container in the fridge 3-4 days. Or, freeze brownies in a freezer bag up to 2 months.