Step 1

Combine orange juice, lime juice, 1 1/2 tablespoons salt, oregano, black pepper, cumin, and garlic in a blender, and process on high speed until smooth, about 30 seconds. Place chicken in a gallon-size ziplock plastic bag, and pour marinade over chicken. Seal and refrigerate, turning occasionally, 8 to 12 hours.

Step 2

Combine red onion slices and 1 cup hot water in a medium bowl; stir 30 seconds. Drain and rinse onions in a strainer with cold running water. Return onions to bowl, and stir in vinegar, 1/2 cup cold water, sugar, and salt until well combined. Set aside.

Step 3

Heat one chimney of charcoal until flames subside and coals are lightly covered in ash. Arrange coals for two-zone cooking: Bank coals along half of bottom grate of charcoal grill. Add oak or mesquite log along edge of coals. Place oiled cooking grate over coals; open bottom and top vents of grill. Place 2 stacks of 3 bricks on grill grate straddling the coals. (Stack bricks so that top brick is 8 to 10 inches above grill grate.)

Step 4

Spray grill basket with cooking spray. Remove chicken from marinade, and discard marinade. Place chicken, cavity side up, in grill basket. Lock basket, and place edges of basket on top of bricks, chicken cavity side down. Grill, flipping basket every 5 minutes, for 30 minutes. Coals will begin to shrink, and chicken will begin to brown. After 30 minutes, remove 1 brick from each stack, and place basket back on bricks. Grill, flipping basket every 5 minutes to expose both the skin side and cavity side to the coals, until chicken is deep brown and an instant-read thermometer inserted in thickest part of thigh registers 165°F, 50 to 55 minutes. Transfer chicken, skin side up, to a rimmed baking sheet, and let rest 10 minutes.

Step 5

Place tomatoes, onion, garlic, and chile on oiled grill grate directly over coals. Grill, turning occasionally, until charred and just tender, about 10 minutes. Remove and discard charred peels from garlic cloves and outer layer from onion. Transfer charred onion, garlic, and chile to a blender; pulse until finely chopped, about 10 pulses. Add tomatoes, cilantro, and salt, and pulse until mixture is chunky in texture, 3 to 5 pulses.

Step 6

Carve chicken, or pull meat off the bone, and coarsely chop meat and skin on a cutting board, reserving any juices on rimmed baking sheet. Transfer chicken to a serving bowl; pour reserved juices over chicken. Drizzle with additional orange juice, and season with remaining 1/2 teaspoon salt. Serve with warm tortillas, pickled onions, and charred tomato salsa.