Peel the potatoes and grate them lengthwise. Place them in a colander or kitchen towel and squeeze out as much liquid as possible. Combine the potatoes in a bowl with the egg, flour, salt, and pepper. Mix well.
Melt 2 tablespoons of clarified butter in a skillet. Drop a heaping tablespoon of the potato mixture into the sizzling butter. Flatten with a spatula and cook for 2 minutes. Turn, flatten again, and cook for another 2 minutes, until crisp on the outside and golden brown. Move to a plate lined with a paper towel while you finish cooking the pancakes. In the same pan, use the remaining butter to fry the eggs, cooking two at a time, until the yolk is a runny or jammy, as desired. Serve 3 – 4 potato pancakes on a plate with a fried egg on top. Sprinkle with salt and pepper serve hot.
Note: To make 6 tablespoons of clarified butter, slowly melt 8 tablespoons of butter in a small saucepan. Set it aside until the milk solids settle. Spoon off any solids that rise to the top and then carefully pour off the golden liquid, leaving the milky part in the bottom of the pan. Clarified butter has a higher burning temperature than melted butter.
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