Pour the chicken broth into your large pot and bring it to a boil. Add the diced potatoes, carrots, celery, and onion, then season with salt and pepper. Reduce the heat and let everything simmer gently until the vegetables turn fork-tender, about 15 minutes.
While the vegetables simmer, melt butter in a separate saucepan over medium heat. Whisk in the flour and let it cook for about a minute until it smells slightly nutty. This step removes the raw flour taste and sets up your thickener.
Slowly pour in the whole milk while whisking constantly to avoid lumps. Keep stirring until the mixture turns thick and bubbly, which usually takes three to four minutes on medium heat.
Pour the thickened milk mixture into the pot with your simmered vegetables and broth. Stir well so everything blends into one creamy base. Let it simmer together for a couple of minutes to marry the flavors.
Lower the heat and stir in the shredded cheddar cheese a handful at a time, letting each addition melt before adding more. Add the diced cooked chicken and stir until it's heated through and the soup looks glossy and smooth.
Give the soup a taste and adjust salt and pepper as needed. If it feels too thick, splash in a little extra broth or milk until it reaches the consistency you like.