Step 1

Using a skillet over medium-high heat, melt the butter and saute the onions until translucent. This may take up to 5 minutes. Set aside, off the heat, when ready.

Step 2

In a large saucepan, whisk together 1/4 cup of melted butter with the flour, cooking over medium-low heat.

Step 3

Cook until the flour is no longer grainy, stirring continuously - add 1-3 tablespoons of milk to keep the flour from burning. This may take 3-4 minutes.

Step 4

Slowly pour the rest of the milk, whisking continuously, into the flour and butter mix.

Step 5

Stir the chicken stock into the saucepan and bring to a simmer.

Step 6

Cook for about 20 minutes, until the mix has thickened.

Step 7

Add the broccoli, carrots, sauteed onion, and celery to the thickened soup and simmer until the vegetables are fork-tender. This may take up to 20 minutes.

Step 8

Stir in the shredded cheese, mixing until the cheese has melted.

Step 9

Salt and pepper to taste.