Place 1 tablespoon butter in a medium saucepan.
Place pan over medium heat.
When butter has melted and is hot, sauté onion until tender and fragrant. Remove from heat, pour into a bowl and set aside.
To same pan, add 1/4 cup melted butter and flour.
Place back over medium heat.
Cook, whisking to combine, for 3 - 5 minutes.
Stirring constantly, slowly add in the half-and-half (you are making a roux).
Add chicken stock, still whisking all the time.
Reduce heat and simmer for 20 minutes.
Add the broccoli, carrots and onions. Cook over low heat until the veggies are tender for 20 - 25 minutes.
Taste. Add salt and pepper, to taste.
The soup should be thickened by now.
Pour in batches into blender and puree.
Return to pot over low heat and add the grated cheese. Stir until melted and well-blended.
Stir in the nutmeg.
Serve hot in a bowl or bread bowl.