Make flax egg: Mix together ground flax and water and set aside to thicken.
Boil potatoes: Peel potatoes and slice into cubes so they cook faster. Add potatoes to a pot of water and bring to a boil. Cook for about 10-15 minutes, until you can easily pierce the potatoes with a fork. Drain and set aside.
Cook Veggies: Meanwhile, add onions and carrots to a pan over medium heat with a splash of water or oil, and cook until onions turn translucent. Add green onion and rehydrated mushrooms (if using) and season with salt and pepper. Stir and turn off heat.
Mash beans: Add white beans to a large bowl and use a fork or potato masher to mash the beans. They don't have to be completely mashed; leave some chunks for texture.
Mash potatoes: Add potatoes to the bowl and mash, again leaving some chunks for texture.
Combine: Add in veggies, vegan cheese, and flax mixture. Mix until well combined.
Form patties: Mix together plant milk and cornstarch. Form the potato mixture into patties. (I ended up with 8 large patties.) Roll each patty in flour, dip in plant milk mixture, then coat with panko. You may need extra flour or panko, depending on how big your korokke are.
Air-Fry: Brush the korokke with oil. Place in the air fryer, leaving a little space between each korokke so they aren't touching each other. You will probably have to fry in two or three batches, depending on the size of your air fryer.
Air fry at 350°F for 15-20 minutes, or until golden and crispy. Serve with Japanese worcestershire sauce, ketchup, mayonnaise, or plain. These are best right after they're made, but leftovers can be reheated in the air fryer to crisp up.