Preheat the oven to 350 degrees (175 C).
Line an 8-inch (20 cm) square pan with a long sheet of aluminum foil that covers the bottom and reaches up the sides. If it doesn’t reach all the way up and over all four sides, cross another sheet of foil over it, making a large cross with edges that overhang the sides. Grease the bottom and sides of the foil with a bit of butter or non-stick spray.
Melt the butter in a medium saucepan. Add the chocolate pieces and stir constantly over very low heat until the chocolate is melted. Remove from heat and whisk in the cocoa powder until smooth. Add in the eggs one at a time, then stir in the sugar, vanilla, then the flour. Mix in the nuts, if using.
Scrape half of the batter into the prepared pan. Drop one-third of the Dulce de Leche, by heaping spoonfuls, evenly spaced over the brownie batter. Then drag a knife through the dulce de leche mounds to swirl it slightly. Spread the remaining brownie batter over, then drop spoonfuls of the remaining Dulce de Leche in dollops over the top of the brownie batter. Use a knife to swirl the Dulce de Leche slightly, again.
Bake for 35 to 45 minutes. The brownies are done when the center feels just-slightly firm. Remove from the oven and cool completely.