Heat the olive oil in a large pot over medium heat. Once the oil begins to shimmer, add the onion and cook for 3-4 minutes, until semi-translucent. Add the garlic, salt, and pepper and cook for 2-3 minutes more, until the garlic is fragrant.
Add the mashed potatoes, broccoli, vegetable broth, and milk and bring to a boil. Cover and reduce the heat to low. Simmer for 20 minutes, or until the broccoli is tender.
Use an immersion blender to blend the soup until smooth.