In a large bowl, combine the yogurt, vegetable oil, lemon juice, garlic, ginger, tandoori masala, cumin, coriander, paprika, turmeric, cayenne pepper, salt, and black pepper. Mix well to create a marinade.
Add the chicken thighs 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 excess marinade to drip off.
If grilling, place the chicken on the grill and cook for about 6-8 minutes per side, or until the internal temperature reaches 165°F. If baking, place the chicken on a baking sheet lined with parchment paper and bake for 30-35 minutes, or until cooked through.
Once cooked, let the chicken rest for 5 minutes before serving. Garnish with fresh cilantro and serve with lemon wedges on the side.