Heat a nonstick skillet over medium heat and add the peanut oil. Add the chicken breasts and cook 2-3 minutes on each side, sprinkling with salt and pepper. Cook until lightly browned.
Add 1 Tbs soy sauce and cook another 3-4 minutes until cooked through. Cut one in half to check--you're going to slice them up later anyway. If the pan gets to dry, add some water. Let the chicken cool a bit while you make up the salad.
Wash, dry and chop the lettuce and add to a large salad bowl.
Peel and grate the carrot (or use the peeler to cut into thin strips). Peel the cucumber and dice. Drain the water chestnuts. Chop the green onions.
Make the dressing: Mix together the remaining soy sauce (3 Tbs), sesame oil, rice vinegar, chili paste, brown sugar, and olive oil.
Cut the chicken into strips. Toss together the vegetables, chicken strips and salad dressing. Garnish with chow mein noodles if you'd like.