Sauté the Aromatics:

In a spacious pot, warm the butter over medium heat until melted. Toss in the chopped onion and cook for 3-4 minutes, stirring until it turns translucent. Then, add the minced garlic and sauté for an additional 1-2 minutes, stirring constantly to avoid scorching.

Cook the Vegetables:

Incorporate the chopped carrots and potatoes into the pot. Pour in the vegetable broth and bring the contents to a boil. Lower the heat to a simmer and let it cook for 15-20 minutes, or until the vegetables are soft and tender.

Prepare the Broccoli:

While the carrots and potatoes are cooking, steam the broccoli florets in a separate pot or microwave until they are tender but still vibrant green. Set aside.

Make the Cream Base:

In a small bowl, whisk together the flour and 1/2 cup of the milk until smooth. Gradually stir this mixture into the pot with the cooked vegetables, followed by the remaining 1 1/2 cups of milk. Continue to stir over medium heat until the soup thickens slightly, about 5-7 minutes.

Add Cheese and Broccoli:

Slowly add the grated cheddar cheese to the soup, stirring constantly until the cheese is completely melted and the soup is smooth. Add the steamed broccoli florets and stir to combine. Allow the soup to simmer for another 5 minutes to meld all the flavors together.

Adjust Seasoning and Serve:

Sample the soup and modify the seasoning with salt and pepper to your liking. Serve hot with crusty bread or crackers for a complete and comforting meal.