Step 1

Finely grind oregano, fennel, and red pepper flakes in spice mill or with mortar and pestle. Place lamb breasts on a work surface, fat side down, and season with salt. Sprinkle spice mixture over top, then rub garlic into meat, working over entire surface.

Step 2

Mix egg yolk, Pecorino, ricotta, olives, parsley, and lemon juice in a small bowl to combine and spread evenly over both lamb breasts. Working with 1 breast at a time and starting from a short end, roll up so lamb is resting seam side down. Starting at the center, tie each roll with kitchen twine at even intervals.

Step 3

Do Ahead: Lamb can be stuffed 1 day ahead. Cover and chill. Bring to room temperature before cooking.

Step 4

Preheat oven to 300°. Heat oil in a large Dutch oven or other heavy pot over medium-high. Season lamb all over with salt and pepper and cook, turning occasionally, until browned on all sides, 12–15 minutes. Transfer to a plate.

Step 5

Add onion, carrot, shallots, and garlic to same pot; season with salt and pepper. Cook, stirring often, until vegetables are browned around edges, 8–10 minutes. Add wine, scraping up browned bits. Bring to a boil and cook until liquid is almost completely evaporated, about 4 minutes. Add lemon, parsley sprigs, thyme sprigs, bay leaves, peppercorns, and 4 cups cold water. Add lamb back to pot and bring to a simmer; cover and transfer to oven. Cook, turning once, until lamb is tender but not quite falling apart, 2–2½ hours. Transfer lamb to a plate and tent with foil to keep warm.

Step 6

Bring braising liquid to a boil over medium-high heat and cook until reduced by half, 15–20 minutes. Taste and season with more salt, if desired. Remove kitchen twine from lamb and slice ¾" thick. Transfer to a platter and strain braising liquid through a fine-mesh sieve over lamb. Top with parsley leaves.