Wrap tofu in paper towel to remove excess water. Repeat the same process 3 to 4 times.
Finely chop cheese and ham.
In a mixing bowl, break the tofu using your fingers and add the egg. Season with salt and pepper and mix well until mixture looks like scrambled eggs.
Add 1/4 cup panko breadcrumbs and mix well.
Add edamame and cheese and mix well.
Scoop up mixture and make small patties. Sprinkle with panko breadcrumbs and shape again. Sprinkle more breadcrumbs (until the patty clumps without breaking up) and shape one last time.
In a skillet over medium heat, put about 1 cm of oil.
When the oil is hot add patties to the pan and fry both sides until golden brown.
Drain on paper towel to remove excess oil and put on a plate.
Serve with tonkatsu sauce and mayonnaise.
Notes
Keep the leftover patties in a storage container and refrigerate for up to 3 days.