Place potatoes in a large saucepan.
Cover with cold water.
Bring to a boil over medium heat.
Once boiling, reduce heat to medium-low.
Simmer for 20 to 30 minutes or until tender; drain well.
Using a potato masher, mash potatoes.
Add salt and pepper to taste; mix well.
Transfer potatoes to a heat proof bowl; cool completely.
While potatoes are boiling, in a medium bowl, stir ketchup, soy sauce, brown sugar, mirin, Worcestershire sauce, ginger, and garlic together to make Tonkatsu sauce; let flavors blend for at least 1/2 hour.
Heat oil in a frying pan over medium-high heat.
Add onion, stirring for 3 minutes or until softened.
Add ground beef, stirring with a wooden spoon to break up mince, for 5 minutes or until browned.
Add garlic, salt and pepper; stirring for 30 seconds.
Remove from heat; cool for 10 minutes.
Combine potato and meat mixture; mix well.
Shape the mixture into balls about the size of your palm.
Dust hands well with four.
Roll croquettes in flour.
Place breadcrumbs in a shallow dish.
Lightly beat remaining egg in a bowl.
Dip croquettes, 1 at a time, in egg, then roll in breadcrumb mixture, pressing firmly to coat.
Pour enough oil into a large, deep heavy-based frying pan to come 2” up side of pan.
Heat over medium heat until hot.
Cook croquettes, in batches, for 3 to 5 minutes or until golden.
Transfer to a large baking tray lined with paper towel; serve.