Ingredients

• 3 tablespoons achiote paste, divided
• 3 tablespoons vegetable oil, divided
• 1 tablespoon water
• 1 4 1/2-to 5-pound whole turkey breast
• 1/4 cup (scant) loosely packed stemmed dried chiles de árbol or chiles japones (about 12 to 16 chiles)
• 3 cups low-salt chicken broth, divided
• 3 garlic cloves, peeled
• 1 1/2 pounds tomatoes
• 1 medium onion, quartered through core end
• 8 mint leaves, coarsely chopped, plus sprigs for garnish
• 3 tablespoons masa (corn tortilla mix)
• Coarse kosher salt

Instructions

1. Whisk 1 1/2 tablespoons achiote paste, 1 tablespoon oil, and 1 tablespoon water in small bowl. Place turkey in 13x9x2-inch dish. Rub achiote mixture all over turkey. Cover and chill at least 4 hours. DO AHEAD: Can be made 1 day ahead. Keep chilled.
2. Place chiles in small bowl. Add enough hot water to cover by 1 inch. Let soak 30 minutes. Drain; discard soaking liquid. Place chiles in blender with remaining 1 1/2 tablespoons achiote paste and 1/4 cup chicken broth. Puree. Strain, discarding solids in strainer. DO AHEAD: Chile broth can be made 1 day ahead. Cover; chill.
3. For charcoal grill, light 30 briquettes in chimney starter; heat until ash-gray. Remove top rack from grill and place 1 disposable aluminum pan on 1 side of grill. Pour briquettes onto side of grill opposite pan. Return rack to grill.
4. For 2-burner gas grill, remove rack; place 1 disposable aluminum pan on 1 side of grill. Return rack and light grill (medium heat) on side opposite pan.
5. For 3-burner gas grill, remove rack and place 1 disposable aluminum pan in center of barbecue. Return rack and light grill on both sides of pan (not under pan).
6. For all grills, thread garlic on skewer. Place garlic, tomatoes, and onion on grill and cook until brown in spots on all sides, about 2 minutes for garlic and 4 minutes for onion and tomatoes. Transfer to work surface. Peel and core tomatoes. Coarsely chop garlic and onion.
7. Place turkey on rack above pan.
8. Close lid; insert thermometer into hole in lid. Cook turkey until instant-read thermometer inserted into thickest part of breast registers 160°F, about 1 hour 15 minutes, maintaining grill's internal temperature at around 350°F by opening and closing vents, adjusting gas grill's burners, or adding more hot briquettes to charcoal grill. Transfer turkey to work surface; let rest 15 minutes.
9. Meanwhile, heat 2 tablespoons oil in heavy large pot over medium heat. Add garlic and onion; sauté until onion is translucent, about 5 minutes. Add chile broth, chopped mint, and 2 1/2 cups chicken broth. Add tomatoes 1 at a time, squeezing into pot to crush. Bring to boil; reduce heat to medium-low. Simmer with lid slightly ajar for 30 minutes.
10. Whisk remaining 1/4 cup chicken broth and masa in small bowl. Whisk 2 tablespoons masa mixture into sauce; stir until sauce thickens, about 5 minutes, adding more masa mixture if sauce is too thin. Season with coarse salt.
11. Cut turkey 1/4 inch thick. Drizzle some of sauce over, garnish with mint sprigs, and serve, passing more sauce alongside.