Preheat the oven to 425 degrees F (220 degrees C). Put 2 empty, greased ramekins in the oven for 5 minutes to heat up.
Melt butter in a skillet over medium-high heat. Spread hashbrowns in a thin layer in the skillet; cook, stirring as little as possible, until golden and crispy.
Carefully remove hot ramekins from the oven. Divide hashbrowns and carefully press hashbrowns into the bottom and up the sides of each ramekin. Press firmly. Sprinkle sausage evenly over hashbrowns; sprinkle with cheese.
Make a small well in the center of each ramekin. Crack an egg into each well. Season with salt and pepper.
Carefully return ramekins to the oven; bake for 12 to 18 minutes, based upon how you want the yolk. 12 to 14 minutes will leave a runny yolk; at 15 to 18 minutes, the yolk should be jammy to firm.