Heat 1 teaspoon olive oil in a large saucepan over medium heat. Add remaining 2 cloves garlic, 1 shallot, and plum tomatoes. Cook, stirring occasionally, for 2 minutes. Add wine and cook until liquid is reduced by half, about 4 minutes. Add duck stock and thyme and let simmer for 20 minutes. Strain liquid through a fine mesh strainer set over another large saucepan, discarding solids. Place meatballs in the liquid and simmer over medium heat, until cooked through, about 30 minutes.
Preparing the Whole Dish:
Heat remaining teaspoon olive oil in a large skillet over medium heat. Add remaining 2 shallots and cook, stirring occasionally, until translucent, about 2 minutes. Add meatballs, cavatelli, mint, cooking liquid, and butter; let cook for 2 minutes.
To plate, place 5 meatballs, some broth and cavatelli in a shallow bowl. Top with grated parmesan cheese and a raw quail egg.