In large bowl, whisk lime juice, vinegar, soy sauce, sesame oil, fi sh sauce, sugar and garlic until sugar is dissolved. Drizzle in oil while whisking constantly until marinade is emulsified. Add chicken leg quarters, cover and refrigerate to marinate for 4 hours.
Remove chicken from marinade; discard marinade. In large sauté pan, sear chicken in oil 3-4 minutes or until browned. Transfer skin side up to a wire rack over a sheet tray. Bake at 350°F for 45 minutes or until internal temperature reaches 165°F.
For The Quickles
In small saucepan, heat vinegar, water, sugar, salt, peppercorns, garlic cloves, and bay leaves to a simmer. In medium bowl, toss cucumber, carrots and jalapeño. Pour brine mixture over vegetables. Cool, cover and refrigerate.
For The Dipping Sauce:
In small bowl, whisk all ingredients. Cover and refrigerate.
Arrange chicken on a large serving platter along with rice, lettuce leaves, tomatoes and quickles. Serve with dipping sauce.