In a large pot, heat 1 tablespoon butter over medium heat. Add onion and cook, stirring frequently, until translucent, about 5 minutes. Add garlic and cook until fragrant, about 30 seconds. Add remaining 4 tablespoons of butter and flour. Whisk to combine the butter and flour into a paste and cook for 1 minute. Slowly add milk, whisking the entire time. Add chicken broth. Bring to a boil, cover, and low to a simmer. Simmer for about 10 minutes or until thickened. Add broccoli and carrots and continue simmering until veggies have softened, about 15 minutes. Remove from heat and stir in cheese until melted. Taste for seasoning and add salt and pepper as needed.