Heat olive oil in a large nonstick skillet over medium-high heat.
Add chopped garlic and rosemary; sauté until fragrant, about 30 seconds.
Stir in ground beef and cook, breaking it apart, until no longer pink about 3 minutes.
Add ketchup, Worcestershire sauce, Dijon mustard, and pepper; stir and cook for 1 minute until thickened.
Divide beef mixture evenly on bottom halves of toasted slider buns.
Top each with shredded Cheddar, baby arugula, and spinach, then sandwich with the top bun.
Serve immediately while warm.
