Peel and cut the potatoes into bite-sized pieces. Place them in a bowl, add 1/2 tbsp water, cover with plastic wrap, and microwave at 500W for 5 minutes. Mash until smooth.

Add salt, pepper, and mayonnaise to the mashed potatoes. Gradually add milk while mixing until the mixture reaches a manageable consistency.

Place a portion of the potato mixture on a slice of ham, add a quarter slice of cheese, top with more potato mixture, and sandwich with another slice of ham.

In a bowl, beat the egg, then mix in the flour and water to create a batter. Coat the ham and potato sandwiches in the batter, then cover with breadcrumbs.