Season the chicken breast with a pinch of salt and half of the minced ginger.
Heat a non-stick skillet over medium-high heat with a tiny drop of oil and sear the chicken for 5-6 minutes per side until golden and cooked through.
Bring a small pot of water to a boil and cook the egg for 6.5 minutes for a jammy yolk, then immediately transfer to an ice bath, peel, and slice in half.
Place the cucumber on a cutting board and lightly smash it with a heavy knife or rolling pin to break the skin, then chop into bite-sized pieces.
In a small bowl, whisk together the soy sauce, sesame oil, rice vinegar, honey, minced garlic, and the remaining ginger.
Toss the smashed cucumber with half of the dressing in a bowl.
Slice the seared chicken and arrange it on a plate with the cucumber salad and the halved egg, drizzling the remaining dressing over the chicken and egg before serving.