DO AHEAD; CAN BE MADE UP TO 3 DAYS AHEAD! Prepare grill to medium high heat. Arrange onion rounds on baking sheet. Brush with oil; sprinkle with 3/4 teaspoon coarse salt and pepper. Transfer onion rounds to grill rack; close cover. Cook until onions are tender, about 10 minutes. Transfer to a medium bowl. Toss with vinegar. Cover and chill.
DO AHEAD; CAN BE MADE UP TO 3 DAYS AHEAD!
Mix ketchup, chilies, adobo sauce, and 2 teaspoons vinegar in small bowl. Season with salt and more vinegar, if desired. Cover and chill.
Shape beef into 6 1/2 inch-thick patties. Sprinkle patties on both sides with coarse salt and pepper.
Prepare grill to medium high heat. Place burgers on grill. Close cover; cook burgers until bottoms start to darken and juices rise to surface, about 3 minutes. Turn burgers; cook to desired doneness, about 3 minutes longer for medium-rare. Top with onions and cheese. Close cover; cook until cheese melts.
Place grilled English muffin or bun bottoms on plates; spread with ketchup. Top with burgers, tomatoes, spinach, and grilled muffin tops. Serve, passing remaining ketchup separately.