In a large bowl, combine 1/2 cup Greek yogurt, lemon juice, cumin, coriander, turmeric, garam masala, smoked paprika, 1/2 teaspoon salt, and cayenne pepper. Add the chicken cubes and mix well to coat. Cover and marinate in the refrigerator for at least 1 hour, preferably overnight.
Preheat the grill or grill pan to medium-high heat. Thread the marinated chicken pieces onto skewers and brush with 1 tablespoon vegetable oil.
Grill the chicken skewers for 4-5 minutes per side until cooked through and slightly charred. Remove from grill and set aside.
In a large skillet, heat the remaining 1 tablespoon vegetable oil over medium heat. Add the chopped onion and cook, stirring frequently, until soft and golden, about 7-8 minutes.
Add the minced garlic and grated ginger to the skillet and cook for 1-2 minutes until fragrant.
Stir in the crushed tomatoes and remaining 1/2 teaspoon salt. Simmer the sauce for 10 minutes, stirring occasionally.
Transfer the sauce to a blender and blend until smooth. Return the sauce to the skillet over low heat.
Whisk the remaining 1/2 cup Greek yogurt, cottage cheese, and heavy cream together in a bowl until smooth. Slowly stir this mixture into the sauce, heating gently but not boiling.
Add the grilled chicken pieces to the sauce and simmer for 5 minutes to combine flavors and heat through.
Garnish with chopped fresh cilantro before serving.