Step 1

Slice pork crosswise into 4 medallions. Place between 2 sheets of plastic wrap; gently pound with a meat mallet to 5"-wide pieces.

Step 2

Whisk koji, salt, and 1 cup water in a medium bowl until salt dissolves. Add pork; cover and chill at least 8–24 hours.

Step 3

Drain pork (no need to rinse off koji if it sticks); set aside. Place cabbage in a medium bowl. Finely grate lemon zest over; slice lemons in half and squeeze in the juice. Season cabbage with salt. Using your hands, toss and squeeze cabbage until slightly wilted, about 5 minutes. Chill until ready to use.

Step 4

Place eggs, panko, and flour in 3 separate shallow bowls. Whisk 2 Tbsp. water into eggs. Line a rimmed baking sheet with paper towels and set a wire rack inside; place near stovetop. Working with 1 piece of reserved pork at a time, dredge in flour, shaking off excess. Dip in eggs, letting excess drip back into bowl, then coat in panko, pressing to adhere before shaking off excess. Transfer to another rimmed baking sheet or a platter.

Step 5

Pour oil into a medium heavy pot to come 2" up sides. Heat oil over medium-high until an instant-read thermometer registers 350°–365°. Carefully lower a piece of pork vertically into skillet, letting it slide away from you into oil until it eventually lies flat. Cook, using tongs to keep meat submerged in oil, until golden brown underneath, about 1 minute. Using tongs, carefully turn over and cook until golden brown on the other side, about 1 minute. Transfer to prepared baking sheet; immediately season with salt. Repeat with remaining pieces of pork.

Step 6

Drizzle tonkatsu sauce over cutlets. Spread mustard on 1 side of half of bread slices; top with cabbage mixture, then cutlet, sauce side up. Close sandwiches and slice each into thirds.