Preheat oven to 375°F. To prepare Braciole, coat a 3-quart rectangular baking dish with nonstick cooking spray; set aside. Trim fat from beef. Cut roast crosswise into 12 slices, each about ½ inch thick. Place one slice of the beef between two pieces of plastic wrap; gently pound with the flat side of a meat mallet to about ⅛-inch thickness. Repeat with the remaining beef slices; set aside. Season meat with ¼ teaspoon salt and ¼ teaspoon pepper.
In a medium bowl, combine chopped eggs, Romano cheese, garlic, parsley, oregano, ¼ teaspoon salt, and ¼ teaspoon pepper. Spoon a scant ¼ cup of the egg mixture onto each beef slice. Roll up. Secure each with a wooden toothpick.
Pour the water into prepared baking dish. Arrange beef roll-ups in dish. Cover dish with foil. Bake about 1¼ hours or until beef roll-ups are tender.
To prepare Basil Tomato Sauce: In a 5- to 7-quart Dutch oven, heat olive oil over medium heat. Add onion and carrot; cook about 10 minutes or until very tender, stirring occasionally. Add garlic and oregano; cook 2 minutes more, stirring frequently. Add dry red wine; bring to boiling, stirring up any browned bits on bottom of the pan. Boil gently, uncovered, about 10 minutes or until most of the wine evaporates. Add tomatoes, salt, and black pepper. Bring to boiling; reduce heat. Cover; simmer for 25 minutes. Uncover and simmer for 20 minutes more. Stir in basil just before serving.
To serve, spoon some of the warmed sauce onto 12 dinner plates. Using a slotted spoon, remove beef roll-ups from baking dish; place roll-ups on sauce on plates. Drizzle roll-ups with a little of the cooking liquid in baking dish. Top with additional warmed sauce.
Tip: To hard-cook eggs, place 6 eggs in a single layer in a large saucepan (do not stack eggs). Add enough cold water to cover the eggs by 1 inch. Bring to a rapid boil over high heat (water will have large rapidly breaking bubbles). Remove from heat; cover and let stand for 15 minutes. Drain. Run cold water over the eggs or place them in ice water until cool enough to handle; drain. To peel eggs, gently tap each egg on the countertop. Roll the egg between the palms of your hands. Peel off eggshell, starting at the large end.