Place the potatoes in a large saucepan and cover them with cold water. Add a pinch of salt and bring to a boil. Cook for approximately 12–15 minutes, or until the potatoes are very tender when pierced with a fork.
Place the sliced cucumber in a bowl and sprinkle lightly with salt. Let it sit for about 5–10 minutes, then gently squeeze out excess moisture.
Prepare the carrot and onion. For a milder onion flavor, soak the sliced onion in cold water for several minutes and drain well.
Mix mayonnaise, rice vinegar, sugar, salt, and black pepper in a small bowl. Add the dressing to the potato mixture and gently fold until combined.
If using chopped boiled egg, fold it in carefully. Refrigerate the potato salad while preparing the chicken.
Combine soy sauce, mirin, honey or brown sugar, rice vinegar, ginger, garlic, and sesame oil in a small bowl.
Pat the chicken dry and cut it into bite-sized pieces.
Heat cooking oil in a large skillet over medium-high heat. Add the chicken and cook for approximately 5–7 minutes, turning occasionally, until browned and cooked through.
Pour the teriyaki sauce into the skillet. Reduce the heat slightly and simmer for 2–4 minutes, allowing the sauce to become glossy and coat the chicken.
Spoon the chilled or slightly cool potato salad onto serving plates or bowls. Arrange the hot teriyaki chicken alongside or over the potatoes.
Finish with toasted sesame seeds and sliced green onions. Add a small sprinkle of shichimi togarashi if desired.