1. Heat a grill to medium.
2. Cut lamb in half following a natural seam that runs across center meat. Brush lamb lightly with 1/3 cup olive oil and rub each side with 1 tablespoon thyme, 1/2 teaspoon allspice, and a generous amount salt. Place lamb on grill fat side up and grill until well browned, about 15 minutes. Turn lamb and grill until other side is well browned and an instant-read thermometer inserted in thickest part registers 130degreesF to 135degreesF for medium-rare, about 10 minutes. Transfer lamb to a cutting board, cover loosely with foil, and let rest for 10 minutes.
3. Whisk 2 tablespoons olive oil, brown sugar, sherry vinegar, ginger, and 1/2 teaspoon salt in a medium bowl until sugar dissolves. Add scallions, papaya, and bell and habanero peppers. Toss to coat with dressing.
4. Slice lamb and arrange on a serving platter or individual plates. Sprinkle lamb with salt to taste and serve with salsa.
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