In a large pot or dutch oven, bring the chicken broth to a simmer and add broccoli and carrots. Simmer for 15 minutes.
While the broccoli and carrots simmer prepare the cream sauce. Melt the butter in a skillet over medium heat. Add onions and saute for about 5 minutes, adding the garlic during the last minute. Add the flour and cook for a minute. Gradually pour in the half and half, stirring constantly. Continue cooking and stirring until the mixture thickens.
Add cream mixture to the broccoli and carrots. Stir in the cheeses, one at a time allowing each to melt before adding the next. Last, sprinkle with black pepper. Be careful not to let the soup boil once you start adding the cheese or it will break down.