Make 4 deep slits evenly spaced on the top side of each chicken leg quarter to help the marinade penetrate.
In a large shallow dish or large resealable bag, whisk together 3/4 cup Greek yogurt, 1/4 cup vegetable oil, 2 tablespoons lemon juice, 1 tablespoon garam masala, 2 tsp ground cumin, 2 tsp chili powder, 2 tsp smoked paprika, 1 1/2 tsp ground coriander, 1 1/2 tsp ground ginger, 1 1/2 tsp garlic powder, 1 1/2 tsp salt, 1/2 tsp turmeric, 1/2 tsp onion powder, 1/2–1 tsp cayenne, and 1 tsp red food coloring (optional) until smooth.
Remove 1 1/2 tablespoons of the prepared marinade and set it aside in a small bowl for the cilantro-yogurt dip.
Add the chicken pieces to the remaining marinade, turning to coat thoroughly. Cover or seal and refrigerate to marinate for 8–24 hours.
About 30–60 minutes before cooking, remove the chicken from the refrigerator to bring it closer to room temperature.
If grilling: preheat the grill to high. Let excess marinade drip off or gently dab so the chicken is coated but not dripping. Sear the chicken on the hot grill for 5 minutes, flip and sear the other side 5 minutes.
After searing, cover the grill and reduce heat so the closed-lid temperature stays about 350°F. Grill with the lid closed for 12 minutes, flip, then grill another 12 minutes. Check with a thermometer; the chicken is done at 165°F and continue cooking as needed.
If baking: preheat the oven to 425°F. Place the chicken on a rimmed baking sheet lined with lightly greased foil and bake 30–35 minutes or until the thickest part reaches 165°F. Broil briefly if you want more golden color.
Make the cilantro-yogurt dip by whisking the reserved 1 1/2 tablespoons marinade with 3/4 cup plain Greek yogurt, 2 tablespoons finely chopped cilantro, and 1 teaspoon lemon juice. Cover and refrigerate until serving.