To make the dressing, whisk the lime juice, fish sauce, dark brown sugar, Thai chilies, sesame oil and vegetable oil together in a small bowl, until the brown sugar dissolves. Set aside.
For the duck:
Season the duck breasts on both sides with salt and pepper.
Heat a large cast iron skillet over medium heat, do not add any oil to the skillet, plenty of fat will render off the duck as it cooks.
Set duck breasts down on a cutting board, skin side up. (The skin side is the white, fatty side.) Using a sharp knife, gently score the skin crosswise in ½” intervals, being careful not to cut into the meat. Rotate the breast and score again, to make a crisscross pattern. This will help the fat to render as the duck cooks.
Heat a large cast iron skillet over medium heat. Place the duck, skin side down in the skillet. Let the duck cook, untouched, until the skin is golden brown and crisp, about 5 minutes.
Turn the duck over and sear the other side for 5 minutes. Turn it once more, skin side down, and sear for an additional 2 minutes. When you remove it, the skin should be crisp but the middle spongy but firm, cooked to a medium rare. (If you are using an instant read thermometer, cook to an internal temperature of 140°F).
Remove the duck from the pan and allow to rest on a cutting board for 5 minutes.
Slice the duck into strips, cutting any large strips into bite size pieces if desired.
For the salad:
In a large bowl, toss together the romaine, apple, pineapple, red onion, green onion, carrot, cilantro and mint leaves. Refrigerate until needed.
When ready to serve, pour the dressing over the lettuce mixture and toss to combine.
Divide the salad among bowls then sprinkle the cashews over the top.
Top with pieces of the crispy duck and serve immediately.