Preparing the Hash-Browns

1. Prep the Potatoes: After peeling and rinsing the potatoes, shred them using a box grater or food processor. Immediately soak the shredded potatoes in a bowl of cold water to prevent browning and remove excess starch.
2. Dry the Potatoes: Drain and rinse the potatoes to remove the starch. Spread them on a clean kitchen towel or paper towels to dry thoroughly.
3. Cook the Potatoes: Heat 2 tablespoons of butter in a large skillet over medium heat. Add the dried potatoes, spreading them out evenly. Season with salt, pepper, and cayenne. Cook without stirring for about 5 minutes to let the bottom layer become golden and crispy. Stir and flip the potatoes every few minutes, adding more butter as needed, until all sides are crispy and golden brown.

Making the Gravy

1. Make the Roux: In a small saucepan, melt 2 tablespoons of butter over medium heat. Stir in 2 tablespoons of flour and cook, stirring constantly, for 1-2 minutes.
2. Add the Stock: Gradually whisk in the stock, ensuring no lumps form. Bring to a simmer and cook until the gravy thickens, about 5 minutes. Season with salt and pepper. Keep warm.

Frying the Eggs

1. Fry the Eggs: In the same skillet used for potatoes, add a little butter and crack the eggs into the pan. Cook to your desired doneness.

Assemble the Dish

1. Serve: Divide the hash-browns among plates. Spoon gravy over each portion and top with a fried egg. Sprinkle with chopped chives. Serve with a grilled slice of bread on the side.