Cook and dry the potatoes. Put the peeled potato chunks in a pot, cover with cold water, and bring to a boil. Simmer until fork-tender, roughly 15–20 minutes. Drain thoroughly, return to the warm pot for about 1 minute to release excess moisture, then mash. Stir in the scallions, parsley, salt, and pepper. Drier potatoes help the croquettes keep their shape.
Cook the shrimp filling. While the potatoes cook, warm about half the 1 tablespoon neutral cooking oil in a skillet over medium heat. Add the minced garlic and stir for about 20 seconds. Add the chopped shrimp and cook, stirring, until the pieces are firm and opaque, about 2–4 minutes depending on size. Stir in 1 teaspoon of soy sauce and set aside to cool. Keep raw shrimp and its utensils separate from ready-to-eat ingredients; cook shrimp until opaque.
Cook the mushroom filling. Heat the remaining cooking oil in the same cleaned skillet. Add the mushrooms and cook for about 4–6 minutes until their released liquid evaporates. Stir in the cooked edamame and remaining 1 teaspoon soy sauce; remove from heat and let cool. Drain and mash the canned sardines separately. Cook off mushroom moisture so the potato mixture does not become soggy.
Divide and flavor the potato mixture. Divide the warm mashed potatoes evenly among three bowls. Fold the sardines into one bowl, the cooled shrimp mixture into the second, and the mushroom-edamame mixture into the third. Spread each mixture out slightly and cool for about 15 minutes, refrigerating if needed, until firm enough to shape. Keep the three varieties separate so their fillings are easy to identify.
Shape the croquettes. Divide each flavored mixture into four equal portions, making 12 croquettes total. With clean hands, form rounded, slightly flattened balls about 2–2 1/2 inches (5–6 cm) across. Place them on a tray. Evenly sized croquettes brown more consistently.
Bread the croquettes. Set out three shallow bowls: flour, beaten eggs, and panko. Coat each croquette lightly in flour, then egg, then panko, pressing the crumbs on gently. Set the coated croquettes on the tray while heating the oil. A single thorough coating is usually enough; work gently so the soft potato does not crack.
Fry in batches. Pour neutral oil into a stable deep, heavy-bottomed pot, leaving the pot less than half full, and heat to 340°F (170°C). Carefully lower in a few croquettes at a time without crowding. Fry approximately 3–4 minutes per batch, turning if needed, until evenly golden and heated through. Transfer with a slotted spoon to a rack or paper towels. Let the oil return to temperature between batches. Do not leave hot oil unattended; lower food gently to avoid splashing.
Serve warm. Let the croquettes sit for a minute or two, then serve three per person on lettuce, if using. Offer tonkatsu sauce or another dipping sauce separately. Tell guests which croquettes contain sardines, shrimp, or mushrooms and edamame. The fillings are different; label them if serving guests with food allergies.
