Prepare the eggs: Bring a large saucepan of water to a simmer over medium-high heat. Add 4 eggs and simmer 5 minutes for very runny yolks, or 6 minutes for yolks that are slightly set around the edges. Transfer the eggs to a bowl of ice water with a slotted spoon; drain, peel carefully and pat dry. Season with salt.
Put the flour in a shallow dish and season with salt and pepper. Put the breadcrumbs in another dish; season with salt and pepper. Beat the remaining 2 eggs in a third dish; season with salt and pepper. One at a time, carefully dredge the cooked eggs in flour, then roll in the beaten eggs, letting the excess drip off, and roll in the breadcrumbs. Roll in the beaten eggs again and reroll in the breadcrumbs. Put the eggs on a parchment-lined plate; refrigerate while you make the gravy. Reserve 2 tablespoons of the seasoned flour.
Make the gravy: Heat the olive oil in a medium skillet over medium heat. Crumble the sausage into the skillet and cook, breaking it up with a wooden spoon, until browned, 3 to 4 minutes. Add the thyme and the reserved 2 tablespoons seasoned flour and cook, stirring, 1 minute. Stir in the chicken broth and cream and bring to a simmer. Cook, stirring occasionally, until the gravy thickens, 4 to 5 minutes. Stir in the parsley and season with salt and pepper. Add more cream or water if the gravy is too thick. Keep warm.
Deep-fry the eggs: Heat about 2 inches of vegetable oil in a medium saucepan over medium heat until a deep-fry thermometer registers 375 degrees F. Using a slotted spoon, carefully lower the breaded eggs into the hot oil and fry, turning, until golden brown, about 2 minutes. Drain on paper towels.