Preheat the oven to 350 degrees F. Grease an 8-by-8-inch baking pan (a square pan will result in thicker brownies).
In a medium-large saucepan, melt the butter with the chocolate over low heat, whisking occasionally until smooth and melted. Sprinkle in the cocoa powder and whisk to combine. Allow to cool for about 10 minutes.
Stir in the granulated sugar and vanilla until just combined. Stir in the eggs. Gently stir in the flour until halfway incorporated. Pour the batter (it will be very thick!) into the prepared pan.
Place the dulce de leche in a glass bowl. Warm it in the microwave for 30 to 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 batter. Don't drag the knife sideways through the batter; hold it so the thin side of the knife leads the way.
Bake for 40 minutes, then check the brownies with a toothpick. If they are overly gooey/messy, return to the oven for 5 to 10 more minutes (just be careful not to burn the edges). Allow to cool completely (important!), then sprinkle with powdered sugar and cut into small squares.
When warm, the brownies will seem overly gooey/soft. As they cool, they will firm up. (Brownies are best several hours after baking, or even the next day!)