Gently beat the egg without creating bubbles.
In a separate bowl, mix dashi, mirin, soy sauce, salt, and sake.
Combine with the egg, then strain for a smooth texture.
Lightly season the chicken with sake and salt.
Divide chicken and shimeji between two heatproof cups.
Pour in the egg mixture. Let sit for 3–5 minutes to release air bubbles.
Place cups in a pot and add water halfway up the sides.
Heat gently and keep the water at 80–90°C (176–194°F). Do not boil.
Cover with the lid slightly ajar and cook for 12–15 minutes.
When just set and softly jiggly in the center, remove from heat.