Heat the olive oil in a large pot over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes. Stir in the minced garlic and cook for another minute.

Add the diced potatoes and vegetable broth to the pot. Bring the mixture to a boil, then reduce the heat to a simmer. Cover and cook for about 15-20 minutes, or until the potatoes are tender.

Once the potatoes are cooked, use a potato masher or a whisk to partially mash the potatoes, leaving some chunks for texture.

Stir in the broccoli florets and continue to simmer for another 5 minutes until the broccoli is tender.

Reduce the heat to low and add the shredded cheddar cheese and milk. Stir until the cheese is melted and the soup is creamy. Season with salt and pepper to taste.

Serve hot, garnished with optional toppings like bacon bits, sliced green onions, and additional cheese.

For a thicker soup, you can blend some of the soup in a blender and then return it to the pot.

Feel free to substitute or add other vegetables such as carrots or cauliflower for a variation.