Step 1Peel the potatoes. Cut each into 4 pieces and rinse them. Cut the carrot in half lengthwise and then crosswise to make 4 pieces. Place the potatoes and carrot in a saucepan of cold water and bring to a boil. After 4 minutes, remove the carrot. Continue boiling the potatoes until tender, about 15 minutes. Drain and return the potatoes to the pot. Cover and shake the pot over medium heat to let the moisture in the potato evaporate. Remove the potatoes from the heat and immediately mash them with salt and pepper to taste.
Step 2Slice the carrot into thin pieces, add to the potatoes and let them stand until room temperature. Meanwhile, peel and thinly slice the cucumber. If using a Japanese cucumber, do not peel it. Sprinkle the cucumber with some salt and mix it with your fingers. When the cucumber is soft, rinse it off, then drain it in a strainer and squeeze out as much water as possible. Slice the onion, then salt and drain in the same way.
Step 3Cut the ham into small pieces. Peel the apple and slice it into small pieces. When the potato mixture has become room temperature, mix in the mayonnaise, optional mustard to taste and salt and pepper to taste. Mix in the onion, cucumber, ham and apple.
Step 4Line each plate with a lettuce leaf and serve the potato salad on top. Pour on tonkatsu sauce to taste.