In a large bowl, combine the yogurt, vegetable oil, lemon juice, garlic, ginger, tandoori masala, cumin, paprika, turmeric, salt, and black pepper. 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 4 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 excess marinade to drip off. Discard the remaining marinade.
Grill the chicken for about 25-30 minutes, turning occasionally, until the internal temperature reaches 165°F and the chicken is charred and cooked through. If using an oven, place the chicken on a baking sheet and bake for 30-35 minutes, turning halfway through.
Once cooked, remove the chicken from the grill or oven and let it rest for 5 minutes before serving.
Garnish with fresh cilantro and serve with lemon wedges on the side.