In a bowl, combine the Greek yogurt, tandoori masala, garlic paste, ginger paste, oil, Kashmiri chili powder, smoked paprika, lemon juice, and salt. Stir until the marinade is smooth and evenly blended.
Place the chicken breasts in the slow cooker and coat them evenly with the prepared marinade, turning the chicken so it is fully covered.
Cover and cook on low for about 3 to 4 hours, until the chicken is fully cooked and tender.
Using two forks, shred the chicken directly in the slow cooker, mixing it gently with the cooking juices and marinade.
In a separate bowl, combine the cream cheese, sour cream, mayonnaise, mozzarella, cheddar, garam masala, paprika, sugar, and tomato paste. Stir until the mixture is smooth and evenly blended.
Add the creamy mixture to the slow cooker and stir well so it mixes evenly with the shredded chicken.
Cover and cook on low for another 20 to 30 minutes, until the cheeses melt and the dip becomes thick and creamy. Stir once during cooking so the dip blends smoothly.
Sprinkle chopped cilantro and sliced green onions over the dip before serving. Serve warm with naan chips, toasted pita wedges, crackers, tortilla chips, or fresh vegetable sticks.