Pre-heat grill, grill pan or large nonstick skillet.
Combine the meat with the grated onion, Worcestershire sauce, tomato paste, hot sauce and steak seasoning. Form four large patties and drizzle each with a touch of EVOO. Grill 4-5 minutes on each side and reserve until ready to assemble.
To make each burger, place the cheddar cheese on a bottom piece of bread, top it with the cooked burger and place the Monterey Jack cheese on top of that. Cover with the remaining piece of bread and put back on the grill pan to melt the cheeses and toast the bread.
Cover the sandwiches loosely with a sheet of tin foil and continue cooking until cheese is melted. Serve with thick cut french fries and mixed green salad.