In a large mixing bowl, add the 1" piece of chicken, Greek yogurt, 2 tablespoons of the olive oil, lime juice, and minced garlic. Use tongs to mix the ingredients and coat the chicken in the marinade. Cover with plastic wrap and transfer to the fridge to marinate for about 4 hours.
After the chicken has marinated, add the remaining 1/2 tablespoon of olive oil to a saute pan and heat over medium high heat. Remove the chicken from the marinade and allow it to drain. Add the chicken to the hot pan and cook until golden brown on all sides and cooked through. Set aside.
To make the tikka masala sauce, add the butter to a large saute pan and heat over medium. Once melted, add the onions and cook, stirring occasionally, until translucent and golden (about 5-7 minutes).
Meanwhile, in a small bowl combine the coriander, cumin, cardamom, nutmeg, paprika, cayenne, and ginger. Mix, then add the spice mixture to the cooked onions along with the tomato puree, water, half and half, and salt. Bring the mixture to a boil while stirring.
Reduce to a simmer, then add the cooked chicken and allow to simmer for about 5 minutes.
Top with fresh cilantro and serve warm over cooked basmati rice or with Naan bread. Enjoy!