Place the minced beef in a bowl, add the salt and black pepper. Mix together with your hands and form the meat into four patties.
Mix the cheeses and squash together into two balls. Place a ball on top of two of the patties, then top with the other pattie. Squash the edges of the patties together firmly. Put to one side.
Brush the insides of the brioche rolls with the oil and place them on the griddle until griddle marks appear (about 45-60 seconds on a hot griddle). Remove from the griddle. Alternatively you can toast under the grill.
Place the burgers on the griddle (or under a hot grill) and cook, turning once until cooked throughout (about 6-8 minutes).
Now it’s time to assemble the burgers. Place the bottom of the burger rolls on plates. Add lettuce and tomato. Top with the burger and red onion slices. Squeeze ketchup on top. Place the burger tops on and serve.