Preheat grill to medium-high.
Sprinkle chicken with ¼ teaspoon each pepper and salt.
Oil the grill rack. Grill peach halves, cut-side down, until lightly charred, 4 to 5 minutes. Transfer to a cutting board and slice.
Grill the chicken until an instant-read thermometer inserted in the thickest part registers 160 degrees F, 3 to 5 minutes. Flip and top the chicken with the peach slices and cheese.
Grill until an instant-read thermometer inserted in the thickest part registers 165 degrees F and the cheese is melted, 1 to 2 minutes more.
Whisk oil, vinegar, tarragon and the remaining ¼ teaspoon pepper and ⅛ teaspoon salt in a large bowl. Add arugula and toss to coat.
Serve the chicken on top of the arugula. Garnish with more tarragon, if desired.
