Put the potatoes in a medium pot and cover with cold water. Add 1 tablespoon of the vinegar and 1 tablespoon salt. Bring to a boil over medium-high heat and continue to boil until tender, about 15 minutes. Drain well and cool.
Whisk the remaining 2 tablespoons vinegar with the Dijon mustard, 1/2 teaspoon salt, 1/4 teaspoon black pepper and the sugar in a small bowl. Pour in the olive oil and whisk until combined.
Spread the potatoes in a single layer on a large serving plate and top with the pieces of hearts of palm. Scatter the piquillo peppers over top and drizzle with the dressing. Sprinkle with the spring onions and salt and pepper to taste.