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1
Roast bell pepper on rack of a gas burner over high heat, turning with tongs, until skin is blackened, 12 to 15 minutes.
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2
(Or broil pepper on a broiler pan about 5 inches from heat, turning occasionally, about 15 minutes.)
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3
Transfer to a bowl and cover tightly, then let stand 20 minutes.
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4
When cool enough to handle, peel pepper, discarding stem and seeds, and coarsely chop.
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5
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.
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6
Peel onions and trim root ends slightly, leaving onions whole, then halve lengthwise and cut halves lengthwise into 1/2-inch-thick wedges.
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7
Insert 1 wooden pick through each wedge to hold layers together while grilling, then put onions in a bowl.
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8
Heat vinegar, sugar, and salt in a small nonreactive heavy saucepan over moderate heat, stirring, until sugar is dissolved.
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9
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.
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10
Prepare grill for cooking.
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11
If using a charcoal grill, open vents on bottom of grill, then light charcoal.
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12
Charcoal fire is medium-hot when you can hold your hand 5 inches above rack for 3 to 4 seconds.
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13
If using a gas grill, preheat burners on high, covered, 10 minutes, then reduce heat to moderate.
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14
Grill onions on a lightly oiled grill rack, covered only if using a gas grill, turning over once, until tender, about 5 minutes total.
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15
Remove and discard wooden picks.
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16
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.
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17
Meanwhile, grill English muffins, turning over once with tongs, until toasted, about 3 minutes total.
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18
Serve burgers on muffins topped with sauce and onions.
