In a medium pot over medium heat, slowly bring the chicken and 3 cups water to a simmer. Once the water begins to simmer, reduce the heat to low and cook until the juices run clear when thickest part of chicken is pierced with the tip of a knife, 10 to 12 minutes. Transfer the chicken to a cutting board, let cool slightly, then cut the chicken crosswise into thin slices.
Meanwhile, make the dressing: in a medium bowl, whisk together the almond butter, 3 tablespoons of the avocado oil, the chili garlic sauce, 2 tablespoons of the rice vinegar, and the soy sauce, toasted sesame oil, and 3/4 teaspoon salt, 1/2 teaspoon pepper, and 3 tablespoons water. (For a thinner dressing, whisk in another tablespoon of water.) Set aside.
In a large bowl, combine the cucumber and lettuce. Add the remaining 2 teaspoons avocado oil and 1 teaspoon rice vinegar and a few pinches of salt and pepper. Gently toss to combine, then distribute the salad between 4 plates or bowls.
In the same large bowl, add the sliced chicken and dressing, toss to combine, then divide it among the plates of salad. Top each with one quarter of the red pepper, scallions, and almonds. Garnish with the cilantro leaves and mint and serve.