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1
Prepare grill for cooking.
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2
If using a charcoal grill, open vents on bottom of grill, then light charcoal.
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3
Charcoal fire is medium-hot when you can hold your hand 5 inches above rack for 3 to 4 seconds.
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4
If using a gas grill, preheat burners on high, covered, 10 minutes, then reduce heat to moderately high.
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5
While grill heats, whisk together yogurt, minced mint, lemon juice, and salt to taste.
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6
Mince 1 half of garlic clove and whisk into yogurt sauce.
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7
Mix lamb, parsley, onion, salt, pepper, and allspice with your hands in a bowl until just combined (do not overwork mixture or patties will be tough).
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8
Form into 4 (4 1/2-inch) patties (1/2 inch thick).
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9
Brush both sides of bread slices with 2 tablespoons oil (total for all slices) and grill, covered only if using a gas grill, turning over once, until golden, 1 to 2 minutes total.
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10
Rub 1 side of each toast with cut side of remaining garlic and season with salt.
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11
Grill patties on lightly oiled grill rack, covered only if using a gas grill, turning over once, until browned but still slightly pink in center, 5 to 7 minutes total.
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12
Toss together mesclun, whole mint leaves, remaining teaspoon oil, and salt and pepper to taste.
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13
Divide lamb among toasts on plates, then spoon sauce over and top with greens.
