Peel the potatoes, then cut into 3cm chunks. Halve the carrot lengthwise, then thinly slice. Put the potato in a medium saucepan of cold salted water and bring to the boil. Reduce the heat to medium and cook for 10 mins. Add the carrot and cook for 2-3 mins until the potato and carrot are tender. Drain, return to the pan and cover to keep warm.
Step 2 — Prep ingredients
Meanwhile, bring a small saucepan of water to the boil for the eggs. Peel and finely grate the ginger. Thinly slice the cucumber. Thinly slice the spring onion on an angle, keeping the white and green parts separate. Put the cucumber and white part of the spring onion in a sieve, add ½ tsp salt and toss to coat. Set aside to drain.
Step 3 — Prep chicken and boil egg
Combine the mirin, ginger and 1 tbs soy sauce in a shallow dish. Add the chicken and turn to coat. Cook 1 egg in the pan of boiling water for 6 mins for soft-boiled, or 8 mins for hard-boiled. Drain, then cool in cold water.
Step 4 — Cook chicken
Heat 2 tsp oil in a medium frypan over medium heat. Remove the chicken from the marinade and reserve the marinade. Cook the chicken, turning regularly, for 10-12 mins until browned and cooked through. Remove from the pan and rest for 3 mins.
Step 5 — Make glaze
Add the reserved marinade, 2 tsp honey and 1½ tbs water to the pan, scraping to incorporate the flavour from the base of the pan. Increase the heat to high and cook, for 1-2 mins until the sauce is reduced to a sticky glaze. Remove the pan from the heat. Add the chicken to the glaze and turn to coat.
Step 6 — Get ready to serve
Peel and halve the egg. Coarsely mash the potato and carrot with a fork. Stir cucumber mixture, aioli, rice wine vinegar and sesame oil into the mash. Taste, then season with salt and pepper. Divide the potato salad, chicken and egg among plates and drizzle the glaze over the chicken. Scatter with sesame seeds and remaining spring onion to serve.