Cook the Vegetables:
In a large pot or Dutch oven, melt the butter over medium heat.
Add the diced onion and sauté until it becomes soft and translucent, about 4-5 minutes.
Add the minced garlic and cook for another 1-2 minutes until fragrant.
Make the Roux:
Sprinkle the flour over the onion and garlic mixture, stirring constantly. Cook for 2-3 minutes, allowing the flour to cook but not brown.
Add the Liquid:
Slowly pour in the chicken or vegetable broth, whisking constantly to prevent lumps.
Gradually add the half-and-half or milk, continuing to whisk until the mixture is smooth and thickened.
Cook the Vegetables:
Add the broccoli florets and diced carrots to the pot.
Bring the mixture to a simmer, reduce the heat to low, and cook for 15-20 minutes until the vegetables are tender.
Blend (Optional):
For a smoother soup, you can use an immersion blender to puree some or all of the soup. Alternatively, you can transfer half of the soup to a blender, blend until smooth, and return it to the pot.
Add the Cheese:
Stir in the shredded cheddar cheese, a handful at a time, allowing it to melt completely before adding more.
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