Make the tonkatsu sauce: In a small bowl, mix together the ketchup, Worcestershire sauce, and sugar until combined. Set aside.
Make the sandwich: Heat 2 inches of canola oil in a medium pot over medium-high heat until it reaches 350˚F (180˚C).
Slice through the thick part of the chicken thigh so it an even thickness. Season with salt and pepper on both sides.
Add the flour, egg, and panko to 3 separate shallow dishes.
Coat the chicken thigh in the flour, then the egg, then the panko.
To test the oil temperature, add a pinch of panko—if it sizzles immediately, it’s ready for frying. Carefully fry the chicken for about 2 minutes per each side, or until golden brown. Drain on paper towels.
In a small bowl, mix together the mayonnaise and hot mustard.
Assemble the sandwiches: Spread the mayonnaise sauce on each piece of toasted bread. On one side, layer on the cabbage, a drizzle of tonkatsu sauce, the chicken, and more tonkatsu sauce. Top with the other slice of bread.
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