In a pot on medium high heat add carrots, broccoli, onion, garlic, chicken stock, and bay leaf. Bring to a boil and cook for 15 minutes, stirring occasionally.
In large pot melt the butter. Add flour one tablespoon at a time while constantly whisking. Cook for a minute.
Remove bay leaf from veggie mixture then add it to the pot with the roux. Stir to combine. Add the half & half and put on low heat. Mix together.
Add all of the seasonings and mix well to combine.
Add gouda and mix until cheese is melted. Follow that with the sharp cheddar and mix until melted.