Season chicken with salt and pepper.
In a small bowl, whisk together vinegar, oil, mustard, tarragon, shallot, and garlic. Place chicken in a large resealable plastic bag. Pour half of vinaigrette mixture over chicken. Seal bag and refrigerate for 2 hours to overnight. Cover and refrigerate remaining vinaigrette.
In a large bowl, combine spring mix, Bibb lettuce, frisée lettuce, beets, Broccolini, and radishes; toss to combine.
Grill chicken, covered with grill lid, over medium-high heat (350° to 400°) for 15 to 20 minutes, or until a meat thermometer reaches 165°. Cool for 5 minutes. Slice chicken, and arrange on salad. Drizzle with remaining vinaigrette; add cheese. Garnish with boiled egg wedges, if desired. Serve immediately.