Step 1: Heat olive oil in a large Dutch oven or heavy-bottomed pot over medium heat. Add diced onion and cook until softened and translucent, about 5-7 minutes.
Step 2: Add minced garlic, diced green chiles, ground cumin, chili powder, and dried oregano to the pot. Sauté for 2-3 minutes, stirring constantly, until fragrant.
Step 3: Pour in the chicken broth, diced tomatoes (undrained), and enchilada sauce. Bring the mixture to a gentle simmer, scraping up any browned bits from the bottom of the pot.
Step 4: Stir in the shredded cooked chicken. Reduce heat to low, cover, and let the soup simmer for at least 15-20 minutes to allow the flavors to meld and deepen.
Step 5: In a small bowl, whisk together the softened cream cheese and sour cream until smooth. To prevent curdling, temper the mixture by adding a ladleful of hot soup to it, whisking until combined, then slowly stir this tempered mixture back into the pot of soup.
Step 6: Stir until the cream cheese is fully melted and incorporated, creating a creamy texture. Season the soup with salt and freshly ground black pepper to taste.
Step 7: Add the shredded Monterey Jack or cheddar cheese to the soup. Stir continuously until the cheese is fully melted and forms a smooth, cheesy consistency. Serve immediately, garnished with your favorite toppings such as fresh cilantro, diced avocado, and tortilla strips.