Place 4 eggs in a small saucepan; add cold water to cover. Bring to a boil; remove from heat, cover, and let stand for 3 minutes. Carefully drain, then fill pan with ice water to cool eggs. Gently crack shells and carefully peel under cold running water. Place eggs in a bowl of cold water; cover and chill until cold. DO AHEAD: Can be made 1 day ahead. Keep chilled.

Place flour in a wide shallow bowl and crushed corn flakes in another wide shallow bowl. Divide sausage into 4 equal portions. Pat 1 portion of sausage into a thin patty over the length of your palm. Lay 1 soft-boiled egg on top of sausage and wrap sausage around egg, sealing to completely enclose. Repeat with remaining sausage and eggs.

Whisk remaining 2 eggs in a medium bowl to blend. Working gently with 1 sausage-wrapped egg at a time, dip eggs into flour, shaking off excess, then coat in egg wash. Roll in corn flakes to coat. DO AHEAD: Can be made 1 day ahead. Keep refrigerated, uncovered.

Attach a deep-fry thermometer to side of a large heavy pot. Pour in oil to a depth of 2” and heat over medium heat to 375°. Fry eggs, turning occasionally and maintaining oil temperature of 350°, until sausage is cooked through and breading is golden brown and crisp, 5–6 minutes. Use a slotted spoon to transfer eggs to paper towels to drain. Season lightly with salt and pepper. Serve warm with mustard.