Heat the olive oil in a large nonstick or enameled cast-iron saucepan set over medium heat. Add the onion, carrots and celery. Cook, stirring occasionally, until the onion is translucent, 5 to 7 minutes.
Stir in the garlic, thyme, garlic powder, salt and pepper. Cook for 1 minute.
Add the potatoes, bay leaves and vegetable broth. Bring to a boil, then reduce the heat so that the broth is simmering. Cook, stirring occasionally, for 10 minutes. Remove and discard the bay leaves.
Place the flour in a medium-sized bowl. While whisking, slowly pour in the milk. Whisk until the mixture is thoroughly combined.
Stir the milk-flour mixture into the soup, along with the broccoli florets. Return to a simmer. Cook, stirring occasionally, until the soup thickens, about 15 minutes.
Remove from the heat and add the Cheddar cheese. Stir until the cheese is melted and combined.
Taste the soup. Season with additional salt and pepper, if desired. Serve.
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Notes
Note 1: Russet potatoes can be substitute for the Yukon Golds. If using russets, I suggest peeling them before using.Note 2: For a non-vegetarian version, feel free to substitute with vegetable broth. The sodium content varies depending on the brand of vegetable broth you choose. I used the Kroger Simple Truth vegetable broth.Please note that I am not a medical or nutritional professional. I provide nutritional information for my recipes as a courtesy to my readers. It is calculated using the LoseIt! calculator. While I attempt to provide information that is as accurate as possible, you should calculate the nutritional information independently before relying on it.