In a large mixing bowl, combine the yogurt, vegetable oil, lemon juice, garlic, ginger, tandoori masala, cumin, coriander, paprika, turmeric, cayenne pepper, and salt. Mix well to create a marinade.
Add the chicken pieces to the marinade, ensuring they are well coated. Cover the bowl with plastic wrap and refrigerate for at least 2 hours, or preferably overnight for maximum flavor.
Preheat your grill or oven to 400°F. If using a grill, oil the grates to prevent sticking.
Remove the chicken from the marinade, allowing any excess to drip off. Place the chicken on the grill or a baking sheet if using the oven.
Grill the chicken for about 20-25 minutes, turning occasionally, until the internal temperature reaches 165°F and the chicken is nicely charred. If baking, cook for 30-35 minutes, flipping halfway through.
Once cooked, remove the chicken from the grill or oven and let it rest for 5 minutes.
Garnish with fresh cilantro and serve with lemon wedges on the side.