Heat oil in a deep fryer or large saucepan to 350 degrees F (175 degrees C).

Fill a saucepan with 1/2-inch water; bring to a boil. Carefully place cold eggs into the saucepan. Cover with a lid, reduce heat to medium-high, and cook until egg yolks are soft, about 6 minutes. Remove eggs from hot water and cool under cold running water. Peel and dry eggs on paper towels.

Mix together sausage, mustard, nutmeg, and cayenne pepper in a large bowl. Shape mixture into 6 equal balls.

Lay a piece of plastic wrap on a flat work surface. Place 1 sausage ball onto the center of the plastic wrap, fold the plastic over sausage, and flatten into an 1/8-inch-thick oval. Pull the plastic back and place 1 egg onto the center of flattened sausage. Pick up plastic wrap, moisten fingertips, and press sausage around egg to cover completely, sealing well. Repeat with remaining sausage balls and eggs.

Place flour into a shallow bowl. Beat 2 eggs in a second shallow bowl. Pour bread crumbs into a third bowl.

Dredge sausage-wrapped eggs in flour, dip in beaten eggs, then press in bread crumbs to coat. Gently toss between your hands to remove excess bread crumbs, then place breaded eggs on a plate.

Cook in batches in the preheated oil until golden, 5 to 6 minutes. Transfer Scotch eggs to a wire rack to cool, about 5 minutes.