In a large pot or Dutch oven, heat the olive oil and butter over medium heat. Add the finely chopped onion and sauté until translucent, about 5 minutes. Stir in the minced garlic and grated ginger, cooking for an additional 1-2 minutes until fragrant.

To the onion mixture, add garam masala, ground cumin, ground turmeric, chili powder, and cayenne pepper (if using). Cook the spices with the aromatics for 2-3 minutes, stirring constantly to prevent burning and to allow the spices to toast, enhancing their flavors.

Add the bite-sized chicken pieces to the pot, stirring to coat them evenly with the spice mixture. Cook the chicken until it's no longer pink on the outside, approximately 5-7 minutes.

Pour in the crushed tomatoes and chicken broth, stirring to combine all ingredients. Bring the mixture to a boil, then reduce the heat to low, allowing it to simmer uncovered for about 20 minutes. This simmering process helps meld the flavors together and ensures the chicken becomes tender.

After the soup has simmered, stir in the heavy cream or coconut milk, giving the soup its signature creamy texture. Season with salt and freshly ground black pepper to taste. Let the soup simmer for an additional 5 minutes to heat through.

Ladle the Butter Chicken Curry Soup into bowls, garnishing with freshly chopped cilantro. Serve hot, accompanied by cooked basmati rice or warm naan bread to complete the meal.