Wash and dry the produce. Finely dice the onion, and chop the broccoli florets into small bite-size pieces. Set aside.
In a large pot, melt the butter over medium heat. Add and sautee the onions until soft and translucent, about 4 minutes. Add the flour and continue to cook an additional 2 minutes, stirring continuously.
Next, slowly pour in the half-and-half and chicken broth, while continually whisking to ensure that the mixture is smooth.
Carefully add the broccoli and carrots. Cook over low heat for 25 minutes or until the vegetables are fork-tender. Stir occasionally.
After 25 minutes, add the cheddar cheese and blend until completely melted. Stir in nutmeg, then add salt and pepper to taste. Serve immediately with baguettes.