Cook the Potatoes & Carrot: Peel the potatoes and chop into quarters or about 2 inch chunks. Aim for same sizes so they cook evenly. Place potatoes and whole carrot in a pot and cover with water about 1 cm above the potatoes. Add 1/2 tbsp salt and bring to a boil over high heat. Reduce to medium heat and cook until carrots for 6 minutes or until you can pierce it with a fork with slight resistance. Cook the potatoes until fork-tender, about 12-15 minutes.
Dry & Season the Potatoes: Drain completely and return to the pot over medium-low heat. Stir with a wooden spatula to evaporate excess moisture. While still hot, season with sugar, salt, pepper, and rice vinegar. Lightly mash, leaving some chunks for texture (to your liking). Transfer to a large mixing bowl to cool completely.
Hard Boiled Eggs: Place eggs into a saucepan and add water 1 cm high. Bring to a boil over high heat and then reduce to medium heat. Cover with a lid and cook for 4 minutes. Turn off the heat and keep covered for 7-8 more minutes. Run under cold water, peel the shell and then roughly chop.
Prepare the Vegetables: Thinly slice the cucumber and onion (root to tip) using a mandolin. Massage sliced cucumber and onion with a pinch of salt. Let sit 5–10 minutes. Rinse and squeeze out excess water, then pat dry with a paper towel. Slice the cooked carrot into thin rounds or half-moons (if large). Alternatively, you can julienne the carrots.
Prepare the Bacon or Ham: If using bacon, place bacon on a cold pan and cook over low heat. Wipe away excess oil as it cooks and continue cooking until crispy to your liking. Transfer bacon onto paper towel to remove excess oil. Chop the ham or bacon into strips, about 1/3 inch wide and 1 inch tall.
Combine: Add chopped egg, cucumber, onion, carrot, and ham or bacon to the cooled potatoes. Toss until ingredients are roughly scattered.
Dress the Salad: Add mayo, optional ketchup or sugar, Japanese mustard, and season with salt and pepper. Mix until combined and everything is evenly coated in the mayo. Taste and adjust seasoning and consistency to your liking. Cover and refrigerate for at least 30 minutes before serving.