Boil eggs to desired doneness (6–7 min for soft, 9–10 for hard). Cool in ice water and peel.
Divide sausage into 4 portions and flatten into thin patties. Wrap each egg completely in sausage and seal edges.
Set up a dredging station: flour, beaten egg, and breadcrumbs. Coat each sausage-wrapped egg in flour, then egg, then breadcrumbs.
To bake: Preheat oven to 400°F. Place eggs on parchment-lined sheet, brush lightly with oil, and bake 25–30 minutes until golden.
To fry: Heat oil to 350°F in a deep pan. Fry each egg 4–5 minutes until golden and crisp. Drain on paper towels.
Let rest 5 minutes before serving. Slice and enjoy with mustard or dipping sauce.