Bake the potatoes directly on the center oven rack until tender, approximately 45 minutes to 1 hour.
Allow potatoes to cool before peeling. Grate cooled/peeled potatoes using the large holes of a box grater. Season liberally with salt and pepper.
Grease a 12-cup muffin pan generously with butter.
Scoop 3 to 4 tablespoons of grated potato into each muffin cup. Use your fingers to gently press the potato down the bottom and up the sides of each muffin cup to make a nest.
Bake for 15 to 20 minutes, until brown and slightly crispy. Watch to make sure they do not burn.
Allow the nests to cool in the pan and then crack an egg into each one. Sprinkle with salt and pepper and bake until the whites are set, about 15 minutes
Notes
If preferred, whisk the eggs before adding them to the nests, for a scrambled egg in the middle.