Heat a pot on medium high heat. Add a teaspoon of oil and sear the lamb on all sides to get nice coloring all around. Turn the heat down to medium and add in the shallot, carrot, celery, and garlic; saute for 3 minutes.
Stir in the broth, tomato paste, and rosemary; cover with a lid and bring to a low simmer for 30 to 40 miuntes until the lamb is tender. Remove the lid and continue to simmer so the sauce concentrates in flavor. Season with salt and pepper to taste. Remove the rosemary sprigs before serving.
In the meantime, bring a pot of water to a boil and cook the pasta according to the directions on the package. When the pasta is finished drain well.
Heat a grill or grill pan on medium heat. Drizzle the eggplant with remaning oil on both sides and season with salt and pepper. Grill on both sides until tender. Then cut in half or quarter and add to the lamb.
Toss the pasta in the lamb and portion. Top with burrata seasoned with salt and pepper and drizzled with olive oil. Garnish with parsley. Serve with bread.