Sprinkle chicken with salt and pepper. Warm 1 Tbsp. olive oil and butter in a large skillet over medium-high heat until butter is foaming. Add chicken and cook, turning once, until golden on both sides, about 10 minutes total. Transfer to a plate.
While chicken is cooking, whisk remaining 3 Tbsp. oil, vinegar, mustard, 1/2 tsp. salt and pepper in a small bowl.
Slice chicken thinly across the grain. Place arugula, pears, chicken and any accumulated juices from plate in a mixing bowl. Top with dressing and toss gently with hands to coat. Divide among four dinner plates. Sprinkle each portion with walnuts and cranberries, and serve immediately.