Pat the 4–5 leg quarters (drumsticks and thighs; skin removed) dry with paper towels. Make 4 deep slits evenly spaced on the top side of each piece of chicken.
In a large shallow dish or large Ziploc bag, whisk together the marinade ingredients: 3/4 cup plain Greek yogurt, 1/4 cup Vegetable oil, 2 tablespoons lemon juice, 1 tablespoon garam masala, 2 teaspoons each ground cumin, chili powder, smoked paprika, 1 1/2 teaspoons each ground coriander, ground ginger, garlic powder, salt, 1/2 teaspoon each ground turmeric, onion powder, 1/2–1 teaspoon cayenne pepper, and 1 teaspoon red food coloring (optional).
Remove 1 1/2 tablespoons of the prepared marinade and put it into a small bowl. Add to that bowl the remaining cilantro-yogurt dip ingredients: 3/4 cup plain Greek yogurt, 2 tablespoons finely chopped cilantro, and 1 teaspoon lemon juice. Whisk until smooth, cover, and refrigerate.
Add the chicken to the remaining marinade in the dish or bag. Turn each piece to coat thoroughly. Cover the dish or seal the bag and refrigerate for 8–24 hours.
Remove the marinated chicken from the refrigerator 30–60 minutes before cooking to take the chill off.
To grill: grease and heat the grill to high. Let excess marinade drip off or gently dab the chicken so it is coated but not goopy. Place chicken on the hot grill and sear for 5 minutes. Flip and sear the other side for 5 minutes. Close the lid and reduce the heat so the grill temperature is about 350°F with the lid closed. Grill covered for 12 minutes, flip the chicken, cover and grill another 12 minutes. Check the internal temperature at the thickest part; it should read 165°F. If it has not reached 165°F, continue grilling (flipping as needed) until it does.
To bake (alternative to grilling): Preheat the oven to 425°F. Place the chicken on a rimmed baking sheet lined with lightly greased foil. Bake at 425°F for 30–35 minutes, or until the thickest part of the chicken registers 165°F. Broil briefly if desired to brown the exterior.
Remove the cooked chicken from the grill or oven and let it rest a few minutes. Serve with the refrigerated cilantro yogurt dip.