Place the potatoes in a sauce pan, adding enough water to cover them. Bring to a boil, salt generously, lower the heat, and cover the pan, simmering until tender, about 15-25 minutes. Set aside 1 cup of cooking liquid, draining the rest.
While the potatoes are boiling, begin caramelizing the onions. Heat 2 T of oil in a pan, add the thinly sliced onions, and sauté on low heat for 35-45 minutes until golden brown and caramelized. (Don't rush this step.)
Heat the remaining 1-2 T of oil in the pot you removed the potatoes from. Heat the oil gently and once warm, add the potatoes back to the pot. Mash the potatoes with the oil, adding the reserved cooking liquid as necessary. Taste and season with salt . Mix in the caramelized onions, reserving some to scatter on top before serving.