Mix together mirin, sake, soy sauce, ginger and garlic. Add chicken and marinate for 24 hours.
Remove chicken from marinade. Let drip on wire rack. Lightly coat each chicken piece in potato starch.
Heat oil in a heavy-bottom pot to 350 degrees F and fry chicken in batches for 4 minutes. Remove from oil and let cool slightly. Return chicken to oil and fry for another 2-4 minutes, depending on size.
Rest chicken on paper towel or newspaper to remove excess oil.
For the miso hollandaise...
Whisk together egg yolks, miso, and lemon juice until fluffy.
Melt butter on the stove. Slowly stream into the egg yolk mixture, whisking continuously. Season to taste.
To assemble...
Lightly toast milk bread. Trim crusts if desired. Set aside.
Bring water to a boil in a large pot. Reduce heat. Stir in vinegar. Crack eggs into a fine mesh sieve and transfer individually to a ramekin. Add egg to pot and poach for 3-4 minutes. Remove with a slotted spoon.
Layer cabbage on top of milk bread toast. Top with karaage chicken. Layer with egg and top with miso hollandaise. Garnish with tobiko.
Notes
The secret to making this recipe is not to overcoat your chicken with potato starch. Keep things light!