Add mashed potatoes, beaten egg, flour and 2 tablespoons of the panko bread crumbs in a large bowl and mix with your hands to combine. If your mashed potatoes were very well seasoned you might not want to add too much salt or pepper at this point. 

Line a baking sheet or cookie pan with parchment. Make a bed of panko bread crubs on the parchment and then scoop piles of your mashed potato mixture on top of the panko. You need the layer of panko on the bottom so make sure you scoop onto panko and not just parchment.  Depending on the size of patty that you want, you should be able to get between 3 and 5 scoops. 

Coat the top of each pile of mashed potato mixture with more panko. Use your fingers to mash the panko into the potato mixture. This will be somewhat sticky process. You can put a little oil on your fingers and that will help make sure you don't get too messy. 

Once you've turned your potato mixture into panko coated patties on a sheetpan, put the pan into the freezer for at least one hour or as long as overnight. 

When you're ready to cook, get a small layer of vegetable oil in a pan and put it on medium heat. If you're checking temperatures you should try to be no higher than 350 degrees F (175 C).

Transfer as many patties as will fit in your pan and put the rest back in the freezer. Cook each patty 5 minutes on the first side and then flip. If you like the color then you should cook the second side for another 4 minutes. If the top still seems too lightly browned for your taste, cook 4 minutes and then flip again and cook the first side for another minute or two.  When you've achieved the correct crispy brown color you're looking for, move each patty to a wire rack or paper towel lined baking sheet to drain.

Salt each patty when you move it to drain. 

Continue cooking other patties from the freezer. You may need to add more oil between each batch to keep the bottom of your pan covered.