Prep Work: Combine the seasonings and set aside. Take out the cream cheese and allow it to soften. (See notes for a quick way to do that.) Shred cheddar and Monterey jack cheese from a block and set aside.
Heat butter and olive oil in a large soup pot over medium heat. Add the onions and the peppers. Soften for 4 minutes, then add the garlic and cook for 1 more minute.
Pat the chicken dry and season each side with salt and pepper. Add it to the soup pot along with all remaining ingredients except for the cream cheese and shredded cheese
Bring the soup to a gentle boil and allow the chicken to slowly cook through (a rapid boil will make it tough). About 15-20 minutes. Remove the chicken and use two forks to shred. Return it to the soup.
Reduce heat to low. Stir in the softened cream cheese until combined, then stir in the shredded cheese until melted.
Taste and add any additional seasonings as needed, then serve!