Place the well-scrubbed potatoes in a large dairy pot and fill with cold water to 1 inch above the potatoes. Bring to a boil and stir in 1 tablespoon of salt. Cook uncovered for 30 to 45 minutes, or until a knife pierces through the potatoes.

Meanwhile, heat the olive oil in a skillet over medium heat. Cook the onions until caramelized, approximately 20 minutes, stirring frequently.

Drain the potatoes in a colander and peel. Return the potatoes to the pot and mash them using a ricer, food mill, or masher.

In a saucepan, heat the butter and milk over medium flame, until the butter melts. Remove from heat and stir to incorporate. Drizzle over the potatoes, whisking continuously. Season with salt and pepper to taste. Mix in the onions and Parmesan cheese. Garnish with walnuts and serve warm.