In a small saucepan heat cooking oil and sesame oil over medium-high heat for 1 minute. Cook and stir garlic and ginger in hot oil for 15 seconds. Remove saucepan from heat; stir in the vinegar, soy sauce, and 3 tablespoons water. Cool completely.
Rinse chicken; pat dry. Place in a plastic bag set in a shallow dish. Pour half of the soy mixture over the chicken; reserve remaining soy mixture. Close bag. Marinate in the refrigerator for 4 to 24 hours.
Meanwhile, for dressing, in a small bowl stir together reserved soy mixture, 2 tablespoons water, the jalapeno pepper, and sugar. Cover and chill for 4 to 24 hours.
Drain chicken, discarding marinade. Grill chicken on the lightly greased rack of an uncovered grill directly over medium coals for 12 to 15 minutes or until chicken is tender and no longer pink, turning once. Cut chicken into bite-size strips. Combine the carrot and jicama. To serve, line 4 salad plates with the lettuce. Top with carrot mixture, cucumbers, chicken, mushrooms, green onions, and peanuts. Stir dressing; drizzle 1 tablespoon dressing over each serving. Makes 4 servings.