Heat 2 tablespoons olive oil in a large Dutch oven over medium heat. Add diced onion and cook 4 to 5 minutes until softened. Add garlic and cook 1 more minute until fragrant.
Add cumin, oregano, chili powder, smoked paprika, and cayenne to the pot. Stir constantly for 30 to 45 seconds until the spices are fragrant and slightly darkened.
Pour in both cans of green chiles and the chicken broth. Stir and bring to a gentle simmer over medium-high heat.
Season chicken breasts lightly with salt and pepper and lower them into the simmering broth. Reduce heat to medium-low and simmer for 18 to 20 minutes, flipping once halfway through, until the internal temperature reaches 165°F (74°C).
Remove chicken and shred using two forks. Return shredded chicken to the pot.
Add drained white beans. Use a spoon or masher to crush about one-third of the beans in the pot to thicken the broth naturally.
Reduce heat to low. Add cubed room-temperature cream cheese and heavy cream. Stir for 2 to 3 minutes until the cream cheese melts completely. Season to taste with salt, pepper, and cayenne.
Simmer on low for 5 minutes. Serve hot with your choice of toppings.