Make the brownie pudding situation:
Preheat the oven to 325° Fahrenheit. Lightly butter or spray 3, 6 inch cast iron skillets. Alternatively, you can use a 9x15 Pyrex dish. As a matter of fact, use the pan you have on hand. It’s not that serious.
Melt the unsalted butter and set aside. Sift the cocoa powder and flour. Set aside.
To the bowl of an electric mixer fitted with the paddle attachment, add the eggs and sugar. Beat on medium/high speed until the mixture is very thick and light yellow. This can take about 5-6 minutes. Reduce the speed to low and add the vanilla bean paste (or vanilla seeds) and espresso liqueur. Mix until combined.
Add the sifted cocoa powder and flour mixture. Mix only until moistened. With mixer still on low, slowly pour in the cooled butter and mix until just combined. Remove the bowl from the mixer and fold in the chocolate chips.
Pour the brownie mixture into the prepared skillets (or dish). Place a pan large enough to hold the skillets (or pyrex dish) in the oven. Add the skillets (or dish) to the pan and add enough hot water to come about halfway up the side of the skillets (or dish).
If you’re using cast iron skillets, bake the brownie pudding for 40-45 minutes. If using a pyrex dish, bake for 1 hour. Check every once in a while to ensure there is still enough water coming halfway up the baking dish. If the water has evaporated, add more.
This brownie pudding is ready when a cake tester (or a bamboo skewer) inserted will come out mostly fudgy and gooey, but not wet. Remove from the oven and place on a rack to cool while you make the glaze.
To a small sauce pan over low heat, add the dulce de leche. Allow it to melt slightly and then add the espresso liqueur. Whisk to combine. Allow the dulce de leche to melt completely and bubble slightly, whisking the entire time. Remove from heat. Allow the glaze to cool slightly then drizzle over the brownie pudding(s). Serve this situation with a nice scoop of ice cream.