Tip:It’s important to cook the chicken over moderate heat. If the grill is too hot, the honey in the glaze will burn.
Heat a gas grill to medium. Put the lettuce and cabbage in a large bowl, cover with a damp paper towel, and refrigerate. In a small bowl, whisk the vinegar with 2 Tbs. of the tarragon, 1 tsp. of the mustard, 1 tsp. of the honey, and the shallot. Whisk in 1/3 cup of the oil and season with salt and pepper.
Remove the tenderloins from the chicken breasts and reserve for another use. Put the chicken in a medium bowl and season with salt and a generous amount of pepper. In a small bowl, whisk the remaining 3 Tbs. tarragon, 2 Tbs. mustard, 2 tsp. honey, and 1 Tbs. oil until well combined. Spread this mixture all over the chicken.
Put the chicken on the grill, cover, and cook the chicken on one side until grill marks appear, 5 to 6 min. Flip the chicken over, cover, and continue to cook until the chicken is cooked through, firm, and golden brown all over, another 4 to 6 min. Transfer the chicken to a clean cutting board and let rest for 5 min. Slice the chicken crosswise on an angle into 1/2-inch slices.
Toss the greens with just enough of the dressing to coat well. Season to taste with salt and pepper. Portion the greens among four plates. Arrange the sliced chicken on and around the greens and drizzle with a little of the remaining dressing. Sprinkle with the blue cheese and almonds and serve.