Whisk mayonnaise, 2 tablespoons oil, tarragon, vinegar, minced garlic, and lemon juice in small bowl to blend well. Season aioli with salt and pepper. Place chicken breasts between sheets of waxed paper. Using rolling pin, flatten chicken to even 1/2-inch thickness; sprinkle with salt and pepper. Heat remaining 2 tablespoons oil in heavy large skillet over medium-high heat. Add chicken; saute until brown and cooked through, about 4 minutes per side. Cool chicken, then tear into narrow 3-inch strips. Place chicken in bowl. Mix in 3/4 cup aioli; season with salt and pepper (reserve remaining aioli for another use). Rub 1 side of each toast slice with cut side of halved garlic. Arrange 4 toasts garlic side up. Top each with 1/4 of watercress, 1 fanned-out avocado half, and 1/4 of chicken. Sprinkle with salt and pepper; top with remaining toasts, garlic side down.