Boil the Potatoes: Peel the potatoes and cut them into chunks. Place in a pot, cover with cold water, add salt, and boil for 15-20 minutes until fork-tender. Drain and cool slightly before mashing.

Prepare the Hard-Boiled Eggs: Boil the eggs for 10-12 minutes, then transfer to an ice bath to cool. Peel and chop into small pieces.

Chop the Vegetables: Finely chop the carrots and cucumber or any crunchy vegetables. Set the chopped vegetables aside.

Mix the Base Ingredients: In a bowl, combine mashed potatoes, mayonnaise, sugar, vinegar, and mustard. Stir until smooth and well-blended.

Incorporate the Veggies and Eggs: Fold in the chopped vegetables and hard-boiled eggs into the potato mixture gently.

Taste and Chill: Adjust seasoning with salt or pepper, cover, and refrigerate for at least an hour before serving.