Bring a large saucepan of water to a simmer over medium-high heat.

Add 4 eggs and simmer for 5 minutes for runny yolks or 6 minutes for slightly set yolks.

Transfer eggs to a bowl of ice water, drain, peel carefully, and pat dry.

Season boiled eggs with salt.

Place flour in a shallow dish and season with salt and pepper.

Place breadcrumbs in another dish and season with salt and pepper.

Beat the remaining 2 eggs in a third dish and season with salt and pepper.

Dredge the cooked eggs in flour, then beaten eggs, and then breadcrumbs.

Repeat the beaten egg and breadcrumb coating.

Refrigerate the breaded eggs while preparing the gravy.

Reserve 2 tablespoons of the seasoned flour for the gravy.

Heat olive oil in a medium skillet over medium heat.

Crumble sausage into the skillet and cook until browned, about 3-4 minutes.

Add thyme and reserved seasoned flour; cook, stirring, for 1 minute.

Stir in chicken broth and cream; bring to a simmer.

Cook, stirring occasionally, until the gravy thickens, about 4-5 minutes.

Stir in parsley and season the gravy with salt and pepper.

Add more cream or water to adjust gravy consistency.

Keep the gravy warm.

Heat about 2 inches of vegetable oil in a medium saucepan to 375°F.

Carefully lower breaded eggs into the hot oil and fry, turning, until golden brown (about 2 minutes).

Drain fried eggs on paper towels.

Serve the gravy and fried eggs on warmed biscuits.