I love this dish not only for the novelty factor (though that's what makes it extra fun to serve at a party!) but also, and more than anything, for the perfect sweet and savory combination it offers!
Technique tip: To brown the butter, stir constantly (so the milk solids don't burn) and listen closely for when the sizzling and spitting of the butter stops; that's your cue to take it off the heat.
Swap option: Instead of a mature cheddar, try using a mature Red Leicester or a mature Cheshire cheese.
Position a rack in the middle of the oven and preheat the oven to 350 F. Line a 9- by 13-inch baking pan (at least 2 inches deep) with aluminum foil (it's OK if the whole pan doesn't get covered, just be sure to cover the bottom and long sides), leaving a few inches overhanging the sides.2.
Melt the butter in a small saucepan over low heat. When it has melted completely, increase the heat slightly to bring the butter to a gentle simmer, stirring continuously, then cook, stirring and scraping the bottom and sides of the pan to pick up any milk solids that may be sticking to the pan, until the butter is golden yellow, smells nutty and has stopped making hissing and popping noises, 2 to 4 minutes, depending on the exact heat level and the material of the pan used. Remove from the heat and stir in the chopped chocolate. Set aside to allow the chocolate to melt.3.
Combine the granulated sugar, brown sugar, salt, eggs, vanilla and instant coffee (if using) in the bowl of a stand mixer fitted with the whisk attachment (or combine them in a large bowl and use a handheld mixer). Mix on medium-high speed until thick and fluffy, 7 to 10 minutes.4.
While the stand mixer is busy whipping, combine the flour and cocoa powder in a medium bowl and stir with a whisk to break up any lumps.5.
Reduce the mixer speed to low. Pour in the warm chocolate-butter mixture, then add the flour-cocoa mixture (I spoon in the dry ingredients to gradually introduce them to the batter and decrease the cloud of flour puff that occurs from dumping it all in at once). Mix on medium speed until uniformly combined.6.
Stop the mixer and remove the bowl. Scrape down the sides and bottom of the bowl to ensure that everything has been well-incorporated. Using a flexible spatula, gently fold in the cheddar cheese cubes, then pour the batter into the prepared pan, spreading it evenly.7.
Bake for 15 minutes, then rotate the pan and bake for 15 minutes more, until the brownies look set and are barely firm to the touch. Remove the pan from the oven and let cool slightly (just a few minutes), then sprinkle the cheese pieces over the top, spreading them evenly for a nice pop of cheddar color.8.
Let the brownies cool completely (this is the hard part) to allow them to set. Slide a butter knife around the edges of the pan to loosen them, then use the overhanging foil to lift the brownies from the pan and transfer them to a cutting board. Cut into squares, then enjoy your new favorite brownies. These keep well in an airtight container (with waxed paper between the layers) at room temperature for about a week.