Spread chicken onto a large cutting board. In a medium bowl, stir together blue cheese, dried figs and pistachios.
Spoon 1 to 2 tablespoons filling onto center of each piece of chicken, compacting slightly. Roll chicken around filling to enclose completely. Secure with a toothpick.
Season chicken with salt and pepper. Heat butter and oil in a large, lidded stainless skillet over medium-high heat. Add chicken and cook 4 minutes without turning. Carefully flip over and cook, covered, an additional 4 to 5 minutes, until internal temperature registers 165° F.
Remove chicken to a clean cutting board and remove toothpicks. Slice on a slight diagonal and serve with salad greens, if desired.