Preheat oven to 425 degrees. Season chicken with salt and pepper. On a clean work surface, lay cutlets flat, with smooth sides down. Layer each cutlet with arugula; crumble goat cheese in the center. Starting with the narrow end, roll up chicken tightly; seal with a toothpick.
In a large nonstick ovenproof skillet, heat oil over medium-high heat; swirl to coat bottom of pan. Cook, seam side down, until golden brown, 1 to 2 minutes. Turn chicken.
Transfer skillet to oven. Cook until chicken is opaque throughout, 10 to 12 minutes. Remove toothpicks and slice chicken crosswise before serving.