In a large pot, heat the olive oil over medium heat. Add the chopped onion and sauté for about 5 minutes until it becomes translucent. Stir in the minced garlic and cook for an additional minute until fragrant.

Add the broccoli florets and diced potatoes to the pot. Stir to combine and cook for another 5 minutes.

Carefully add the vegetable broth to the pot, ensuring the vegetables are fully submerged. 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 and broccoli are tender.

Once the vegetables are cooked, use an immersion blender to puree the soup until smooth. If you prefer a chunkier texture, blend only half of the soup and mix it back into the pot.

Stir in the milk and heat the soup over low heat until warmed through. Season with salt and pepper to taste.

Ladle the soup into bowls. If desired, top with grated cheese before serving.