Dice the potatoes and carrots into small pieces. Dice the onion and mince the garlic. Chop the broccoli florets into small, bite-sized pieces. This prep work ensures all your ingredients are ready to add promptly during cooking.

Melt the butter in a dutch oven or large pot over medium-low heat. Add the diced onion and minced garlic, and cook them for 7-10 minutes, stirring occasionally, until the onion is softened and translucent. Taking this step slow draws out maximum flavor from the aromatics—I always find this makes the soup base especially delicious.

Add the diced carrots to the pot with the sautéed onion and garlic. Increase the heat to medium and cook for 5 minutes, stirring occasionally, until the carrots begin to soften.

Add the diced potatoes and vegetable broth to the pot. Raise the heat to bring the mixture to a boil, then reduce to a simmer. Cook until the potatoes are tender and easily pierced with a fork.

In a separate bowl, whisk together the milk and flour until there are no lumps. Pour this mixture into the simmering soup and stir to combine thoroughly. This will help thicken the soup and give it a silky texture.

Add the finely chopped broccoli to the pot. Simmer the soup for 10-15 minutes, stirring occasionally, until the broccoli is tender. I like to chop the broccoli very finely so it cooks quickly and blends into the soup nicely.

Gradually add the freshly grated cheddar cheese to the soup, stirring constantly until it melts completely and the soup thickens. Continue to cook the soup to your desired thickness—the potatoes will start to break down, further thickening the mixture. Taste and add salt as needed before serving.