Bring a large pot of water to a boil. Peel the potatoes and cut them into thirds. Add the potatoes to the boiling water and cook until fork tender- roughly 10 minutes. Remove from heat and let cool.
Shred the potatoes using a cheese grater. Add egg, cornstarch and salt and pepper. Toss until well combined. On a baking tray lined with parchment, place 2 tablespoons of the hasbrown mixture, use your hands to shape into rectangles. Repeat until all the hashbrowns are made. Freeze for 30 minutes. Meanwhile, shape the sausage patties. Use your hands and roll out 2 inch balls, flatten each ball using your hand or the back of a plate just as you would if making pressed burgers. Set aside in the fridge until ready to cook.
Heat some oil in a large pan over medium-high heat. Add enough oil to cover the bottom of the pan. Fry the hashbrowns 4-5 minutes per side until golden.
Air-Fryer: preheat an air-fryer according to directions. Spray the basin with some cooking oil, place the hasbrowns into the basin and coat with more cooking oil. Cook at 400 degrees F for 10 minutes. Flip and cook an additional 10 minutes or until crispy and golden.
Remove some of the excess oil from the pan using paper towel. Cook the sausage patties in the same pan on medium for 10 minutes flipping halfway until fully cooked through. During the last minute of cooking, add sliced cheese on top and let melt. Set aside.
Finally, cook the eggs soft or medium poached for runny yolk and toast the buns.
To serve: spread your choice of sauce on toasted buns, top with crispy hashbrown, sausage patty, cheese and egg. Enjoy!