Divide the ground beef into 8 equal portions (about 3 ounces each). Roll them into balls, being careful not to overwork the meat, and set them on a baking sheet. Set aside.
Heat the grill over medium-high heat (between 400 and 450℉) with a griddle on top.
When the griddle is hot, add 2 tablespoons of oil and the sliced peppers and onions. Sprinkle with a pinch of salt and pepper. Cook, tossing often, until the veggies are tender, about 8 minutes total. When the veggies are cooked through, transnfer them to a plate or bowl and cover with foil to keep warm. Set aside.
Butter and toast buns over medium heat until golden on the buttered side. Transfer to a platter and set aside.
Place 4 burger balls onto the hot griddle. Working quickly, firmly smash straight down into a thin patty. Season the patties with a pinch of salt, pepper, a drizzle of yellow mustard. Cook 2 minutes on the first side, or until seared and juices start to come to the surface.
Scrape under the burger with a sturdy spatula and flip. Cover and cook for 1 more minute. Top half of the burgers with American cheese, and half with the Monterey Jack cheese. Transfer the cooked smashed burgers to a platter and continue until all of the burgers are done.