In a large pot over medium-high heat, add your 3 tbsp butter. Once melted and frothy, add your minced garlic, onion, carrots and broccoli. Sauté until tender, about 10 minutes.
Sprinkle sautéed vegetables with flour, stir until coated. While stirring, add your chicken stock and milk.
Once combined, add in your half and half, cheddar cheese and S&P to taste. Allow soup to come together over medium heat for at least an additional 10 minutes before serving.