Preheat the oven to 325 degrees F. Generously spray an 8-inch square baking pan or an 8 x 10-inch baking pan with nonstick baking spray.
Place chocolate in a microwave-safe bowl and heat it in about 45-second intervals, stirring each time, until it's completely melted. Set it aside to cool for 20 minutes or so.
Add the butter and sugar to the bowl of an electric mixer fitted with the paddle attachment. Cream together until light and fluffy.
Add the eggs, one at a time, beating well after each addition. With the mixer on low, slowly drizzle in the melted chocolate. Mix until combined.
Add the flour and vanilla extract. Mix until combined, scraping down the sides of the bowl halfway through. Spread the batter into the greased pan.
Place the dulce de leche in a glass bowl. Warm it in the microwave for 30-45 seconds. Drop large dollops of dulce de leche on the surface of the brownie batter. Use a knife to slowly swirl it through the brownie batter. Don't drag the knife sideways through the batter, hold it so the thin side of the knife leads the way.
Bake brownies for 45-50 minutes, or until the center is almost totally set. Let the brownies cool in the pan for 15 minutes. Carefully turn them out of the pan and let them cool completely. Cut the brownies into squares and serve. Top with ice cream and extra dulce de leche, if desired.