Step 1

Combine lemon segments, shallot, 1 teaspoon salt, and sugar in a medium bowl. Gently whisk in 1/3 cup olive oil; stir in capers, olives, 2 tablespoons parsley, Aleppo, and cayenne. Set aside.

Step 2

Preheat grill to very high (500°F). Brush salmon with remaining 2 tablespoons olive oil; season with thyme, pepper, lemon zest, remaining 1 tablespoon parsley, and remaining 1 1/2 teaspoons salt. Grill salmon, skin sides down, covered, until skin is golden and crispy, about 3 minutes. Flip fish, and grill, uncovered, to desired degree of doneness, 2 to 3 minutes for medium-rare.

Step 3

Transfer fillets to dinner plates, and top evenly with relish.