Prepare the Veggies: Start by peeling and chopping your potatoes into bite-sized cubes. Chop the onion and mince the garlic, and set them aside.
Sauté the Aromatics: In a large pot, melt the butter over medium heat. Add the chopped onion and minced garlic, sautéing until the onion is translucent and fragrant (about 3-5 minutes).
Cook the Potatoes: Add the diced potatoes to the pot and pour in the vegetable broth. Bring to a boil, then reduce to a simmer for about 10-15 minutes, or until the potatoes are tender.
Add Broccoli: Toss in the broccoli florets and let them cook for an additional 5 minutes.
Blend for Creaminess: For a creamy texture, use an immersion blender (or transfer to a standard blender) to puree part of the soup. Blend until you reach your desired consistency while leaving some chunks for texture.
Mix in the Cheese and Milk: Return the pot to low heat and stir in the shredded cheddar cheese and milk. Continue stirring until the cheese is melted and well combined. Season with salt and pepper to taste.
Serve Up: Ladle the soup into bowls and top with your favorite toppings like crispy bacon bits, fresh herbs, or additional cheese!