1. In a small mixing bowl, mash the avocado with a fork until very smooth. Stir in the mayonnaise and lime juice, and continue stirring until the mixture is smooth. Stir in 1/2 teaspoon of salt and refrigerate, covered, until ready to use.

2. Brush the chicken breast tenders with 1 tablespoon of the olive oil. Sprinkle with the salt and pepper. Brush the cut sides of hamburger buns with the remaining tablespoon of olive oil.

3. Grill the chicken on the greased rack of a covered gas or charcoal grill directly over medium heat for 8 to 10 minutes, turning once, or until chicken is no longer pink (at least 170°F). Place hamburger buns on the grill, cut-side down, during the last few minutes of grilling to toast.

4. Allow chicken to rest 5 minutes. Meanwhile, spread some of the Avocado Aioli on both cut sides of each toasted bun.

5. Coarsely shred the chicken with two forks; then roughly chop the shredded chicken. Place some of the chicken on the bottom half of each of the four buns. Add some of the cucumber, tomato, arugula, and red onion to each, and cover with the burger bun tops. Serve the remaining aioli with the meal so everyone can put more on their burgers if desired.