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1
Preheat the broiler, placing the rack 6 inches below the heating element, or set an outdoor grill to medium.
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2
Butter the English muffins and broil or grill until the butter has melted and the muffins are toasted.
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3
Set aside.
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4
In a medium bowl, combine the turkey, egg, mustard, onions, salt, and pepper and mix lightly.
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5
Form the mixture into 4 patties about 3/4 inch thick (they will be a little sticky.
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6
To help form the patties, handle as little as possible and wet your hands in between forming each patty).
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7
To broil: Lightly oil a broiler pan.
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8
Put the burgers on the pan and broil for 3 to 4 minutes, or until the burgers are light brown and feel firm but still slightly springy to the touch.
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9
Turn, place 2 slices of cheese (or one-quarter of the grated cheese) on each burger, and cook 3 to 4 minutes more, or until the burgers feel firm but still give just a little when pressed (for medium).
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10
Cook about 2 minutes longer for well-done.
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11
To grill: Place the burgers on the grill, cover, and cook 3 to 4 minutes, or until the undersides are brown but not dark.
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12
Turn burgers, place 2 slices of cheese (or one-quarter of the grated cheese) on each burger, cover, and cook 2 to 3 more minutes, or until the burgers feel firm but still give slightly when pressed for medium rare.
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13
Cook about 2 minutes longer for medium or well-done.
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14
Serve with sliced tomatoes and extra Dijon on the side.