Chop the onion, carrots, and celery finely, then mince the garlic. Grate the sharp cheddar and Monterey Jack cheeses. If starting with raw chicken, brown the chicken breasts in olive oil, then shred or dice them; otherwise, have 3 cups of cooked chicken ready.
Melt the butter in a large pot over medium heat. Cook the diced onion, carrots, and celery until they are softened, about 8 to 10 minutes. Add the garlic, thyme, smoked paprika, and dry mustard powder; cook for another 2 minutes until you smell the spices.
Sprinkle the flour over the vegetables and stir constantly for 2 to 3 minutes to remove the raw flour taste. Slowly pour in the chicken broth while whisking to prevent lumps, then add the milk and cream. Bring the mixture to a gentle simmer and season with salt and pepper.
Turn the heat down to low. Add the grated cheddar and Monterey Jack cheeses little by little, stirring until all the cheese has melted smoothly. Avoid boiling the soup after adding the cheese.
Cook the elbow macaroni separately in salted boiling water until just tender, then drain well.
Add the cooked chicken and drained pasta into the soup pot. Stir everything gently and let it simmer gently on the lowest heat for 5 to 10 minutes so the flavors meld beautifully. Taste and adjust seasoning or thickness if needed. Serve hot with your choice of chopped fresh parsley, chives, extra cheese, crispy beef bacon, or croutons on top.
