Place 1/2 cup butter in a 6-Quart pot until melted at medium heat.
Add the 1/2 cup flour and whisk to combine, cooking the flour for 2 minutes.
On a cutting board, dice 1 onion.
Wash the carrots in a sink colander, then peel and shred the carrots using a box grater.
Add the diced onion and shredded carrots to the pot and cook until tender, around 3-5 minutes.
On a cutting board, dice the broccoli and then add the diced broccoli and simmer for 3-5 more minutes.
In a separate bowl mix together the heavy cream and water. When all of the butter is melted and mixed well with the flour, slowly add cream/water mixture.
Bring to a slow boil and allow it to thicken a bit before adding the stock.
Add 4 cups chicken broth to the pot, then bring the mixture to a simmer for 20-25 minutes.
Shred the cheddar cheese while the mixture is simmering.
Add shredded cheese a little at a time with a whisk making sure it blends well with the soup.
When all of the cheese is mixed in, taste and add salt and pepper to taste, and finally serve and enjoy!