Heat olive oil in a large nonstick skillet over medium-high heat. Add chopped garlic and rosemary, sauté for 30 seconds until fragrant.
Add ground beef, breaking it apart with a spoon, and cook until no longer pink (about 3 minutes).
Stir in ketchup, Worcestershire sauce, Dijon mustard, and pepper. Cook for 1 minute until slightly thickened.
Divide the beef mixture among the bottom halves of toasted slider buns.
Sprinkle shredded Cheddar on top of the beef, add arugula and spinach, then place the top halves of buns and serve immediately.
