BEEF FILLING

In large nonstick skillet, heat oil over medium-high heat; sauté garlic and rosemary until fragrant, about 30 seconds. Add beef; cook, breaking up with spoon, until no longer pink, about 3 minutes. Stir in ketchup, Worcestershire sauce, mustard and pepper; cook, stirring, until slightly thickened, about 1 minute.
Fill buns with beef mixture, Cheddar cheese and arugula. Serve with slaw (see method below).

CARROT SLAW

Peel and grate carrot using box grater; set aside. In large bowl, whisk together oil, vinegar, honey, salt and pepper. Stir in coleslaw mix and carrot; let stand for 5 minutes. (Make-ahead: Cover and refrigerate for up to 24 hours.)