Dice the onion small, mince the garlic, and shred the carrots.
For the broccoli, chop 2/3 of it into tiny florets [just cut off as much stem as you can and it'll fall apart in the pot] and then take the remaining 1/3 and give it a rough chop so you have a few large pieces to add to the soup. (optional)
Add your veggie broth to a pot over medium-high heat, add 2/3 of your broccoli [about 2 cups of florets] as well as your onion, garlic, carrots, and bay leaf. Simmer on medium, covered, for about 15-20 minutes or until the vegetables are tender.
Start your roux when the vegetables are almost done: In a large pot, melt 3 tablespoons of butter over medium heat. Once melted, slowly add 3 tablespoons of flour as you whisk constantly. Cook the roux about 2 minutes to remove the flour taste.
Slowly pour the broth from your vegetables into the roux and whisk to incorporate.
Remove the bay leaf from the vegetables. Next, pour the vegetables into the roux/broth mixture and slowly stir in 1 cup of room temperature half and half.
Turn the burner to very low heat.
Season with the above herbs and spices.
Add the cheese in small batches until all melted.
Add salt & pepper to taste.