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Saute onion in 1 tbsp of melted butter until transparent.
In a separate pot, make a roux using the remainder of melted butter and 1/4 cup flour. Cook approx. 5 minutes over medium heat, constantly stirring with a whisk.
Slowly add in half & half, continue stirring.
Add chicken stock, stir and let simmer for 20 minutes.
Add in broccoli florets, carrots, and sauteed onions. Let cook for another 25 minutes over low heat.
Stir in cheese until melted, add nutmeg and salt and pepper to taste.