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