In a large mixing bowl, combine the yogurt, lemon juice, vegetable oil, minced garlic, grated ginger, tandoori spice mix, cumin, paprika, salt, black pepper, and cayenne pepper (if using). 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 any excess to drip off. Place the chicken on the grill or a baking sheet if using the oven.
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 baking, cook for 35-40 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.