In a large pot or Dutch oven, melt the butter over medium heat. Add chopped onion and sauté until soft (about 4–5 minutes).
Sprinkle in the flour and cook for 1–2 minutes, stirring constantly to create a light roux.
Slowly whisk in half-and-half and chicken broth until smooth.
Add the chopped broccoli and carrots. Cover and simmer on medium-low for 20–25 minutes or until veggies are tender.
Season with salt, pepper, and nutmeg (if using).
Blend a portion of the soup for a creamier texture using an immersion blender or a regular blender.
Return the soup to low heat and stir in the shredded cheddar until melted and smooth.
Serve hot with crusty bread or a bread bowl.