For the patties: In a large bowl, gently mix the brisket, salt and a few grinds of black pepper. Make into 4 equal-sized patties.
For the layers: In a cast-iron pan on medium heat, add the butter and then the red onions in 4 piles. Immediately place a patty on each mound of onions and gently press to settle. Cook until the onions char a bit, about 5 minutes. Carefully flip each burger, keeping the onions with them so that they now rest on top. Cover and cook until desired doneness. Transfer the patties to a plate and immediately top with the American cheese first, then the pepper Jack. Cover with aluminum to allow the cheeses to melt.
For the buns: Wipe out the pan and keep it on medium heat. Butter the outsides of the buns, then place them buttered-side down on the pan and cook until toasted, about 2 minutes.
Place a patty on each bottom bun and top with pickles, lettuce, a squirt of mustard and the top bun.