Mix vinegar and sugar in a large bowl. Add shallot and set aside. Boil 1 egg.
Boil potatoes in salted water for 12 minutes. Drain and add to bowl with shallot.
Heat 1 tablespoon of olive oil over medium heat. Add turnips and cook until soft about 4 minutes. Add green beans and cook for 1 minute. Add garlic and cook for 30 seconds until fragrant. Add 2 tablespoons of water to the pan. Stir and cook until water is evaporated. Sprinkle in salt and pepper to taste. Mix turnip mixture to potatoes. Let cool for at least 5 minutes.
Add chopped egg, vegenaise, 2 tablespoons of olive oil. Stir and serve! If you like, you can chill for a couple of hours if you prefer cold potato salad.