Preheat grill to medium.

Brush onion slices with evoo and place on heated grill and cook until dark brown on both sides. Add pastrami slices to grill to crisp up, be sure to turn often to prevent burning.

Divide meat into 4 even amounts. Loosely form meat into hamburger patties about 4 ½ inches wide and 1 ½ inches thick. Make an indentation with your thumb in the center of each patty (this will keep the patty from puffing up as the meat gets hot and creates steam). Season patties with evoo, salt and pepper. 

Grill burgers, on each side about 5-7 minutes for medium-rare. While burgers are grill, brush buns or challah with evoo and lightly toast buns on grill. 

Place a burger on each bun half, or lettuce leaf, top with pastrami, a pile of onions, avocado slices, tomato slice and ketchup and mustard.

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Preheat broiler to low.

Broil onion slices for about 3-5 minutes, stirring frequently until browned. Add pastrami slices and continue cooking for another 5 minutes.

Broil patties on each side for 3 minutes for medium rare.