Prep your ingredients. Dice the onion, carrots, and celery. Chop the broccoli into small, even florets.
Shred or dice your cooked chicken. Shred the cheddar. This makes the cooking flow smoothly.
Sauté the aromatics. Warm the olive oil or butter in a large pot over medium heat.
Add onion, carrots, and celery with a pinch of salt. Cook 5–6 minutes, stirring, until softened. Add garlic and cook 30 seconds, until fragrant.
Build the base. Sprinkle in the flour and stir to coat the vegetables.
Cook 1 minute to remove the raw flour taste. If using cornstarch, skip this step and make a slurry later.
Add liquids and broccoli. Slowly pour in the chicken broth while stirring. Add the broccoli, Dijon mustard, smoked paprika, and a little black pepper.
Bring to a gentle simmer and cook 8–10 minutes, until the broccoli is tender but still bright green.
Thicken the soup. If you used flour earlier, the soup should already be slightly thick. If using cornstarch, whisk 2 tablespoons cornstarch with 2 tablespoons cool water to make a slurry, then stir it into the simmering soup and cook 2 minutes, until thickened.
Stir in the chicken. Add the shredded chicken and reduce heat to low. Let it warm through for 2–3 minutes.
Finish with dairy. Stir in the milk and reduce heat to the lowest setting.
Gradually add the shredded cheddar by the handful, stirring until melted before adding more. Remove the pot from heat. Whisk the Greek yogurt in a bowl with a ladle of hot soup to temper it, then stir it back into the pot.
Season with salt and pepper to taste. Add red pepper flakes if you like a little kick.
Adjust and serve. If the soup seems too thick, add a splash of broth or milk. If it’s too thin, simmer a few extra minutes.
Ladle into bowls and top with a little extra cheddar or cracked pepper.