How to Make It
Place the eggs in a pot, then cover with 1 inch salted water. Bring to a boil over high heat; as soon as it is boiling, cover the pot and remove from heat. Allow to sit for 4 minutes in the hot water, then carefully transfer to a bowl of ice water. Allow to cool, about 5 minutes. Carefully peel the eggs.
Combine the sausage, mustard, parsley and nutmeg, mixing together by hand. Divide the sausage mixture into 6 portions; spread each one out into an oval shape. Add an egg to the sausage, then gently form the sausage around the egg as evenly as possible.
Warm the cooking fat to 350° in a skillet, then add an egg or two. When first adding the eggs, gently roll them back and forth to allow the sausage to cook evenly and retain a round shape. Cook until the sausage is cooked through, about 5 minutes per egg.
Preheat oven to 170°. Transfer the cooked egg to a plate lined with a paper towel, then place in the oven to stay warm while finishing the other batches of eggs.
To bake the eggs instead of frying, place the sausage-covered eggs in large muffin tins and bake at 425° until golden and cooked through, about 30 minutes.