Roast bell pepper on rack of a gas burner over high heat, turning with tongs, until skin is blackened, 12 to 15 minutes. (Or broil pepper on a broiler pan about 5 inches from heat, turning occasionally, about 15 minutes.) Transfer to a bowl and cover tightly, then let stand 20 minutes. When cool enough to handle, peel pepper, discarding stem and seeds, and coarsely chop. Puree pepper in a blender with dressing, garlic, paprika, and salt until smooth, then transfer to a bowl and chill, covered, until ready to serve. Peel onions and trim root ends slightly, leaving onions whole, then halve lengthwise and cut halves lengthwise into 1/2-inch-thick wedges. Insert 1 wooden pick through each wedge to hold layers together while grilling, then put onions in a bowl. Heat vinegar, sugar, and salt in a small nonreactive heavy saucepan over moderate heat, stirring, until sugar is dissolved. Pour pickling liquid over onions, stirring occasionally to coat with liquid, and let stand 5 minutes (onions will brighten in color), then drain onions and pat dry. Prepare grill for cooking. If using a charcoal grill, open vents on bottom of grill, then light charcoal. Charcoal fire is medium-hot when you can hold your hand 5 inches above rack for 3 to 4 seconds. If using a gas grill, preheat burners on high, covered, 10 minutes, then reduce heat to moderate. Grill onions on a lightly oiled grill rack, covered only if using a gas grill, turning over once, until tender, about 5 minutes total. Remove and discard wooden picks. Sprinkle salt and pepper on both sides of burgers, then grill burgers on lightly oiled grill rack, covered only if using a gas grill, turning over once with a spatula, 5 to 6 minutes total for medium-rare. Meanwhile, grill English muffins, turning over once with tongs, until toasted, about 3 minutes total. Serve burgers on muffins topped with sauce and onions.