Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium heat.
Add the diced onion and sauté for 5 minutes until translucent and soft.
Add the minced garlic and cook for 60 seconds until fragrant, ensuring the garlic does not burn.
Stir in the ground cumin, dried oregano, chili powder, sea salt, black pepper, and cayenne pepper to toast the spices for 1 minute.
Pour in the chicken bone broth and the diced green chiles (with liquid).
Place the raw chicken breasts into the broth. Bring the mixture to a gentle boil, then reduce heat to low and simmer, covered, for 15-20 minutes or until the internal temperature of the chicken reaches 165°F (74°C).
Remove the chicken from the pot and transfer to a clean board. Use two forks to shred the meat into bite-sized pieces.
While the chicken is being shredded, add the softened cream cheese cubes to the simmering broth. Whisk continuously until the cheese is fully melted and the liquid is smooth.
Stir in the heavy cream and the cauliflower rice (if using). Let simmer for 5 minutes to allow the broth to thicken.
Return the shredded chicken to the pot. Stir well to combine and heat through for an additional 2 minutes.
Taste and adjust salt or pepper as needed. Serve hot, garnished with shredded Monterey Jack cheese, fresh cilantro, or sliced avocado.