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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.
In a large saucepan, whisk together ¼ cup of melted butter with the flour, cooking over medium-low heat.
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.
Slowly pour the rest of the milk, whisking continuously, into the flour and butter mix.
Stir the chicken stock into the saucepan and bring to a simmer.
Cook for about 20 minutes, until the mix has thickened.
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.
Stir in the shredded cheese, mixing until the cheese has melted.
Salt and pepper to taste.