In a medium pot of boiling water, add 4 of the eggs, and cook for exactly 7 1⁄2 minutes. Drain the eggs and transfer to a bowl of ice water to chill. Drain once more, and then peel.
Divide the sausage into 4 balls and then flatten each ball of sausage into a patty. Wrap 1 patty of sausage around each egg and pinch at the seams to completely encase it. Place the flour, remaining 2 eggs, and breadcrumbs in 3 separate shallow dishes, and beat the eggs until smooth. Dredge each sausage-wrapped egg in flour, and then dip in the beaten eggs. Coat the eggs in breadcrumbs, and then transfer to a plate.
Pour oil to a depth of 2 inches in a 6-qt. saucepan and heat until a deep-fry thermometer reaches 350°. Add the eggs to the oil, and fry until golden brown and the sausage is cooked, about 2 minutes.
Using a slotted spoon, transfer the eggs to paper towels to drain, and then season with salt. Halve the eggs lengthwise and serve while hot, with remoulade sauce for dipping.