Step 1

Preheat grill to medium-high.

Step 2

Sprinkle chicken with 1/4 teaspoon each pepper and salt.

Step 3

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.

Step 4

Whisk oil, vinegar, tarragon and the remaining 1/4 teaspoon pepper and 1/8 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.