Peel the broccoli stems and chop the stems and florets into bite-size pieces. In a large pot, bring 4 cups salted water to a boil over medium heat. Add the broccoli and cook until tender, about 10 minutes. Meanwhile, in a skillet, heat 1 tablespoon extra-virgin olive oil over medium heat. Add the onion and potato and cook, stirring, until tender and browned, about 15 minutes. Stir into the pot along with the milk; warm through and season with salt and pepper. Top each serving with 1/4 cup cheese.