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Melt butter over medium heat in a 6-quart Dutch oven or large soup pot. Add onion and cook for 3 to 4 minutes (or until softened) while stirring frequently. Add garlic and cook until fragrant (about 30 seconds), stirring constantly.
Slowly add flour (about a tablespoon at a time) while constantly stirring. Once all of the flour is added and there is a paste formed, slowly pour in milk while continuing to stir. Cook for 1 minute while whisking to smooth out any lumps.
Slowly pour in the chicken broth and stir to combine.
Add broccoli, carrots, salt, pepper, nutmeg, and bay leaf. Stir to combine. Bring to a simmer and simmer for 10 to 15 minutes (stirring occasionally) or until broccoli and carrots are tender. Reduce heat as necessary to maintain a gentle simmer.
Remove from heat and add cheese (a handful at a time) while stirring in between each addition until melted and smooth. Season with additional salt and pepper, if desired. Remove and discard bay leaf before serving.