Saute onions, potatoes, and carrots in 1 Tbsp butter over medium heat in a large pot. Continue to cook until potatoes are tender. Remove from the pot and set aside.
Melt 1/4 cup butter in the same pot. Whisk in flour and cook for 3-5 minutes, whisking constantly to make a roux.
Slowly add half-and-half, whisking constantly until smooth. Add chicken broth and reduce heat.
Simmer on low heat, uncovered, for 10 minutes.
Add broccoli, potatoes, carrots, and onion. Cover and simmer for 25 minutes.
Stir in shredded cheese, then add salt and pepper to taste.