Melt butter in a large pot over medium heat. Add onion and garlic, cooking until softened.
Stir in the flour and cook for 1 minute, stirring constantly.
Gradually add the chicken broth and milk, whisking to combine.
Bring to a simmer and add the broccoli and carrots. Cook until tender, about 15 minutes.
Reduce heat to low and slowly stir in the cheddar cheese until melted.
Season with salt and pepper to taste. Serve warm.