Preheat the oven to 350°F. Line a 9x13-inch baking pan with aluminum foil, leaving an overhang for lifting out the brownies, and lightly grease the foil with melted butter or nonstick spray.

Put the dry chocolate fudge cake mix in a large bowl, then add 1/4 cup of melted butter and the eggs. Stir with a sturdy spoon or spatula until combined and glossy; the batter will be very thick.

Scrape the thick batter into the prepared pan and press or spread it into an even layer, reaching into the corners as best you can.

Bake on the center rack for 18 to 22 minutes, beginning to check at 18 minutes. The top should be set and a toothpick inserted in the center should come out with moist crumbs rather than raw batter.

While the brownies bake, stir the powdered sugar and remaining 1/4 cup melted butter together in a medium bowl. Add 2 tablespoons of hot water a tablespoon at a time, stirring until smooth, glossy, and thick but pourable; add up to 1 more tablespoon of water if needed.

As soon as the brownies come out of the oven, pour the warm frosting over the hot brownie layer and quickly spread it into the corners with a spatula or by gently tilting the pan.

Let the brownies cool in the pan on a wire rack for 30 to 45 minutes, until the frosting is set and the brownies are still slightly warm underneath. Lift them out using the foil overhang, peel away the foil, and cut into 12 squares. Serve warm or at room temperature.