In a large saucepan, over high heat, bring potatoes and enough water to cover to a boil. Reduce heat to low, and simmer 15-20 minutes. Drain potatoes, let cool to room temperature.
In a large saucepan, gently place eggs and fill with water to 1-inch above the eggs. Bring to a boil, then remove from heat and cover tightly for 15 minutes. Transfer eggs with a slotted spoon to a bowl of ice and cold water and let stand 5 minutes to cool. Peel eggs and coarsely chop.
Combine eggs and remaining salad ingredients in a large bowl with a wooden spoon, until well combined. Chill, covered tightly, until serving.
To serve, rinse lettuce of choice and spin dry or dry between paper towels. Arrange one leaf of romaine (Cos) on each salad plate (or divide arugula among the plates), then top with a scoop of the egg and potato salad. Sprinkle some chopped walnuts on top.