Heat the pot: In a large pot, melt 2 tbsp unsalted butter and 1 tbsp olive oil over medium heat until bubbly.
Sauté veggies: Add 1 medium onion, diced, 3 garlic cloves, minced, 2 medium carrots, diced, and 2 celery stalks, diced to the pot. Cook for about 5-7 minutes until the vegetables soften and the onion becomes translucent.
Cook chicken: Incorporate 1 lb boneless, skinless chicken breasts, diced into the pot. Season with salt and pepper to taste and cook for about 5 minutes or until the chicken is no longer pink.
Add flour: Sprinkle 2 tbsp all-purpose flour over the chicken and vegetable mixture. Stir to coat evenly and cook for an additional 1-2 minutes to eliminate the raw flour flavor.
Pour in broth: Gradually stir in 4 cups low-sodium chicken broth until smooth. Increase the heat slightly and let the mixture come to a gentle simmer.
Cream and seasonings: Once the broth is simmering, stir in 2 cups heavy cream, 1 tsp smoked paprika, ½ tsp dried thyme, and ½ tsp ground mustard powder. Let it simmer for about 5 minutes to meld the flavors.
Add broccoli: Add 3 cups fresh broccoli florets, chopped to the pot and cover. Cook for approximately 10-12 minutes until the broccoli is tender yet still vibrant.
Melt the cheese: Lower the heat and incorporate 2 cups shredded sharp cheddar cheese and ¼ cup grated Parmesan cheese (optional). Stir gently until the cheese melts completely, resulting in a creamy soup.
Taste and adjust: Sample the soup and adjust seasoning with additional salt and pepper if necessary.
Serve: Ladle the soup into bowls and garnish with optional croutons, fresh parsley, or extra cheese if desired. Enjoy with bread or crackers on the side to soak up the delicious broth!