Best Ham Soup EVER

Melissa got this recipe from a friend, and it is probably my favorite soup ever! Everyone asks for us to share this recipe.

4 c. water
5 red potatoes, cubed (leave the skins on)
4 carrots, cut into slices
1 onion, chopped
3-4 tsp. salt
1 tsp. pepper
1 bag of frozen broccoli
3 c. cooked ham, cubed
1 stick butter
1/2 c. flour
3 c. milk
4 c. shredded sharp cheddar cheese

Combine the water, salt, pepper and all the vegetables except the frozen broccoli in a big pot. Bring to a boil. About 5 minutes before the vegetables are done, add the broccoli. Cover and boil until all the veggies are done. Meanwhile, in a separate pan, melt the butter over a medium heat. Add the flour slowly, whisking constantly to make it smooth. Add one cup of milk. Continue whisking. Add another cup of milk. Still whisking. Add half the cheese. Don't stop whisking! Add the last cup of milk. I hope you're still whisking! Add the last of the cheese. Keep whisking until the whole mixture is perfectly smooth. Re-attach your arm. Now pour the cheese mixture into the cooked vegetables. DO NOT DRAIN THE WATER from the vegetables - you definitely want this still. Stir until well blended. Turn to low heat and add the cubed ham. Leave on warm heat and stir a bit until the ham is warmed through. Serve and enjoy.