In a large pan begin to melt butter over medium-low heat. Add minced garlic and let cook for about a minute. Whisk in flour and cook for a few minutes, just until golden brown.
Slowly add in chicken stock, half-and-half, kosher salt, white pepper and nutmeg. Stir together and let cook for about 5-10 minutes, stirring occasionally.
Add in the broccoli and carrots and let simmer for another 20 minutes.
Mix in cheese and stir until just combined. Serve in bread bowls or with rolls.