Preheat the grill on medium-high heat.
Heat a large skillet over high heat. Once hot, add the oil, carrot, onion
. Stir to blend
and season with salt. Cook until the vegetables become tender and slightly caramelized, 10-12 minutes. Transfer the vegetables to a bowl and refrigerate to cool them down.
In another bowl, mix the ground brisket
with the mayonnaise. Season with salt to taste. Mix in the vegetables. Form the meat into eight even patties that are about 3/4 inch thick.
Place the hamburgers on the hottest part of the grill
and cook, undisturbed, for about 2 minutes. Turn them a quarter turn and cook for 2 more minutes. Use a metal spatula
to flip them on their second side and cook for an additional 2 minutes for rare to medium rare. Toast the bun halves on the coolest part of the grill. Butter
each one. In the last 2 minutes of cooking, add the cheese on top of each burger to allow it to melt
as the burgers finish cooking. Place a hamburger on a "bottom" bun half and top with some red cabbage
slaw (see recipe below). Top with a bun "top" and serve immediately.