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This super-fast and healthy smothered chicken tastes best on the deck with a chilled glass of gewürztraminer. The fruity white pairs nicely with the peaches here.
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.