Combine cheese and garlic in a small cup. Trim excess fat from chicken; season with salt and pepper. Using a sharp knife, cut a horizontal slit through thickest portion of each breast to form a deep pocket. Stuff each pocket with an equal amount of cheese mixture and arugula. Press edges of chicken together to seal.
Heat oil in a large nonstick skillet over medium-high heat. Brown chicken, 4 minutes per side. Reduce heat to medium; add broth to pan. Cover and simmer, about 6 minutes, or until chicken is cooked through. Serve immediately, discarding any remaining broth.