In a medium sized soup pot melt the butter then sautee the onions for 3-5 minutes. When the onions begin to turn translucent add the garlic and thyme and continue to cook for another minute.
Add the flour and stir continuously for 30-60 seconds to form a roux. Then add the broth, milk, cream, bay leaves and nutmeg. Whisk vigorously til any flour clumps are gone. Bring to a simmer. Taste and adjust with salt and pepper.
Add broccoli and carrots, and cook, covered for another 6-8 minutes.
Remove pot from heat, and stir in cheddar. Adjust seasoning with salt and pepper and serve.