Add the butter, sugar, cocoa powder, and salt to a medium-sized microwave-safe bowl with a handle (such as a large glass measuring bowl). Heat for 1 minute on high to melt the butter. Stir to mix the ingredients.
Add the vanilla and mix. Let the mixture cool for a few minutes while you prepare the pan.
Spray the baking dish with cooking spray. Cut two strips of parchment paper long enough to fit the pan and hang over the sides to act as "handles" to remove the brownies after baking. Place one strip across the pan and press it into the dish. The cooking spray will help adhere the parchment paper to the dish. Place the other strip in the opposite direction.
Add the eggs to the chocolate mixture and stir until well-mixed. Add the flour and stir until just mixed. Be careful not to overmix the batter.
Pour half of the brownie mixture into the pan and spread it with the back of a spoon. The batter will be thick. Add the can of dulce de leche by placing dollops over the chocolate. Use a spoon to spread it. Add the rest of the chocolate mixture, and again spread it with a spoon until all the dulce de leche is covered.
Place the dish in the oven and bake for 40-45 minutes, or until an inserted toothpick comes out with crumbs and a little dulce de leche. If the toothpick is covered in liquid, bake the brownies a little longer. You can also tell if the brownies are cooked by wiggling the pan. If the center is easily moved and liquid-like, cook it longer. The brownies are done when an inserted toothpick comes out mostly clean. Remove them from the oven and let them sit for one hour before cutting so they firm up.
Use the parchment paper handles to remove the brownies from the pan.