Place potatoes into a large pot and cover with salted water. Bring to a boil over high heat, then reduce heat to medium-low, cover, and simmer until tender, about 20 minutes. Drain and allow to steam for a few minutes. Meanwhile, place garlic, olive oil, vegetable broth, and spinach into a blender. Puree until smooth and set aside.
When the potatoes are ready, mash until smooth, then fold in Parmesan cheese, and season to taste with salt and pepper.