Heat a large pot with ⅛ cup salted butter (2 Tbsp). Add in 1 cup chopped carrots, 1 cup chopped celery, and 1 cup chopped yellow onion. Cook 2-3 minutes until veggies soften. Now add 2 minced garlic cloves and cook 30 seconds more.
Pour in the carton of chicken broth and 3 cups chopped yellow potatoes. Bring the pot to a boil then reduce to let simmer until the potatoes begin to soften but are still firm when poked with a fork, about 5 minutes.
Add 3 cups broccoli florets, 1 tablespoon fresh thyme, 1 teaspoon salt, and ½ teaspoon ground black pepper. Cook 3-4 minutes, or until broccoli softens when poked with a fork.
While the soup pot is simmering, in a saucepan melt ¼ cup salted butter with ⅓ cup all-purpose flour. Mix together.
Add 3 ½ cups whole milk and slowly stir until there are minimal lumps and a creamy sauce forms.
Remove the soup pot from heat once the broccoli and potatoes are tender. Pour the creamy liquid from the saucepan along with 3 ½ cups shredded cheddar cheese to the soup pot. Gently mix together until cheese has melted. Ladle into bowls and it’s ready to serve!