In a large bowl, mix together the lime juice, vinegar, tamari, hot sauce and honey. Stream in the canola oil while whisking and season with salt and pepper to make a vinaigrette. Add the cucumber and the remaining slaw mix and stir to combine. Set aside while you prepare the quesadillas.
Place a large skillet over medium heat. Add 1 tortilla to the pan and cook about 1 minute per side, until it starts to blister. Add a quarter of the vegetable and pineapple mixture, a quarter of the shredded chicken and a quarter of the cheese to half of the tortilla and fold the other half up to create a half moon. Cook until cheese is melted, flipping halfway through, and repeat with the remaining tortillas. Cut into wedges and serve with the slaw salad alongside.