In a large saucepan, heat oil over medium; add carrots, onion, and garlic. Season with salt and pepper. Cook, stirring occasionally, until softened, 4 to 5 minutes.
Add tomato paste and cook, stirring, 1 minute. Add turkey; cook, breaking up meat with a spoon, until no longer pink, 4 to 5 minutes.
Add tomatoes, sugar, vinegar, and Worcestershire sauce. Cook, stirring occasionally, until slightly thickened, 12 to 14 minutes. Serve on whole-wheat rolls.