
Cheddar Burgers with Balsamic Onions and Chipotle Ketchup
['1 pound red onions, cut crosswise into 1/3- to 1/2-inch-thick rounds', 'Olive oil', '3/4 teaspoon coarse kosher salt', '1/2 teaspoon coarsely ground black pepper', '2 tablespoons balsamic vinegar', '1 cup ketchup', '1 1/2 teaspoons chopped chipotle chiles from canned chipotles in adobo* plus 2 tablespoons adobo sauce from can', '2 teaspoons (or more) balsamic vinegar', '2 1/4 pounds ground beef (15% to 20% fat)', 'Coarse kosher salt', '6 thick slices sharp cheddar cheese', '6 large English muffins or hamburger buns, split, cut sides grilled', '6 tomato slices (optional)', '2 cups fresh spinach leaves']

Prepare barbecue (medium-high heat): Arrange onion rounds on baking sheet. Brush with oil; sprinkle with 3/4 teaspoon coarse salt and pepper. Transfer onion rounds (still intact) to grill rack; close cover. Cook until grill marks appear, about 4 minutes per side. Reduce heat or move onions to cooler part of grill. Close cover; cook until onions are tender, about 10 minutes. Transfer to medium bowl. Toss with vinegar. DO AHEAD: Can be made 3 days ahead. Cover; chill.
Mix ketchup, chiles, adobo sauce, and 2 teaspoons vinegar in small bowl. Season with salt and more vinegar, if desired. do a h e a d Can be made 3 days ahead. Cover and chill.
Shape beef into six 1/2-inch thick patties. Sprinkle patties on both sides with coarse salt and pepper. Prepare barbecue (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 muffin bottoms on plates; spread with ketchup. Top with burgers, tomatoes, if desired, spinach, and muffin tops. Serve, passing remaining ketchup separately.
*Dried, smoked jalapeños in a spicy tomato sauce called adobo; available at some supermarkets and at specialty foods stores and Latin markets.
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