1. Place potatoes and carrot in a pot, cover with water and bring to a boil. Reduce heat, cover and cook for about 15 minutes, or until fork-tender. Drain and return the pot with potatoes and carrots to the stove top with the burner turned on. Shake the pot for a minute to dry the vegetables out a bit. Cool to a room temperature.
2. To get perfect hard-boiled eggs, place them in a saucepan and cover with cold water. Make sure the tops of the eggs are covered by at least an inch of water. Bring the water to a full boil, cover and remove from the heat. Let stand for 15 minutes. Then place the eggs in a bowl of cold water.
3. Place sliced onion and cucumber in a bowl, add salt and toss. Let stand for about 12 minutes, then firmly squeeze the cucumbers and onions (you can use paper towel) to let some of the juice out.
4. In a small bowl, combine mayonnaise, lemon juice and sugar. Season with pepper.
5. Chop potatoes an carrot into bite size pieces. Chop the eggs.
6. In a large bowl, combine potatoes, carrots and mayo.
7. Add cucumber, onion and eggs, stir to combine. Adjust salt and pepper to taste.
8. Store in refrigerator until serving.