Heat olive oil in a large pot over medium heat. Add onion and cook for 4–5 minutes until soft and translucent.
Add garlic and cook for 30 seconds until fragrant.
Add chicken breasts, chicken broth, green chilies, cumin, chili powder, smoked paprika, oregano, salt, and black pepper.
Bring to a gentle boil, then reduce heat to low. Cover and simmer for 18–22 minutes until chicken is fully cooked.
Remove chicken, shred with two forks, and return it to the pot.
Stir in cream cheese until fully melted and smooth.
Add heavy cream and shredded Monterey Jack cheese. Stir until thick and creamy.
Simmer uncovered for 5–8 minutes, stirring occasionally, until desired consistency is reached.
Taste and adjust seasoning before serving.