Preheat the oven to 350°F and line a 13 X 9 baking sheet with foil. Slice the Hawaiian Rolls in half and place the bottom half in the prepared pan.
Press the ground beef into a large cast iron (or non-stick) skillet. Season with salt and pepper and cook over medium high heat until the meat is crisp and brown (around 4 minutes). Turn the beef and cook for another 3-4 minutes. Remove the beef to a plate lined with a paper towel.
Cook the hash browns until crisp according to package instructions, in the skillet you used for the hamburger meat.
Place the cooked hamburger on the bottom half of the bread in the pan. Top with the cooked hash browns.
Add one teaspoon of the oil and the onions to the skillet. Season with salt and cook over medium heat until soft and translucent.
Scatter the cooked onions over the top of the hash browns
Heat the remaining teaspoon of oil in the skillet. Pour in the well-beaten eggs and lightly scramble. Top with cheese, and then add the top half of the buns.
Brush the tops of the buns with melted butter and bake for 7-8 minutes, until the cheese is melted. Serve warm.