In a large bowl, combine the yogurt, vegetable oil, lemon juice, garlic, ginger, tandoori masala, cumin, paprika, turmeric, salt, black pepper, and cayenne pepper. Mix well to create a marinade.
Add the chicken pieces to the marinade, ensuring they are fully coated. Cover the bowl with plastic wrap and refrigerate for at least 4 hours, preferably overnight for maximum flavor.
Preheat your grill or oven to 425°F. If using a grill, lightly oil the grates to prevent sticking.
Remove the chicken from the marinade, allowing excess marinade to drip off. Discard the remaining marinade.
Place the chicken on the grill or a baking sheet lined with parchment paper if using the oven. Cook for 25-30 minutes, turning halfway through, until the chicken is cooked through and has a nice char. The internal temperature should reach 165°F.
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.