Prepare the hash brown mixture with hot water according to package directions.
While the hash brown mixture rehydrates, slice onions into rings and place in pan with olive oil, black pepper, salt and sugar. Cooker over low heat until caramelized.
Beat 4 eggs until well-beaten. Combine with cheddar cheese.
Once the hash browns are rehydrated, strain them and press out any excess liquid.
Combine hash brown mixture with egg mixture until well-mixed.
Fill a flat-bottomed fry pan with vegetable oil until it is around 1/2 an inch in depth. Heat until it spits when a drop of water is flicked in it. On a gas stove, this will be medium heat.
Place egg rings in oil and then place a 2-inch diameter ball of the hash brown mixture into them. Press down until the 'bun' is flat and fills the ring. Allow to cook for around 30 seconds before pulling out the egg ring.
Cook until they are golden brown at the edges. Carefully flip them to cook on the other side. It should take around 2-3 minutes on each side.
While the last buns are cooking, heat your cast iron skillet and cook your burger patties (shaped by the egg rings as well) until your preferred level of 'doneness'.
When all patties and hash brown buns are cooked assemble with caramelized onions and ketchup.