Combine the peanut oil, ginger, garlic, cilantro, green onion, and red pepper. Add the chicken and marinate in the refrigerator overnight. Remove the chicken from the marinade and season with salt and pepper.
Preheat the oven to 350 degrees F.
Heat a large ovenproof saute pan over medium-high heat. When the pan is hot, add enough peanut oil to coat the bottom of the pan. Add the chicken and sear on both sides. Place the pan in the oven and roast until cooked through, about 10 to 15 minutes. Remove from the oven, cool slightly, and dice into 1/2-inch pieces.
Preheat a grill or grill pan. Drizzle the onion slices with oil and place on the grill. Cook until the onions are soft, about 4 minutes per side. Remove from the grill.
Place the chicken into a bowl and add barbecue sauce, to taste. Place 4 tortillas on a flat surface. In a separate bowl, mix the fontina and mozzarella cheese. Place some of the cheese mixture on each tortilla. Top with some of the chicken and grilled onions. Top each with another tortilla.
Heat a large saute pan over medium-high heat. Add 1 tablespoon of the butter and cook until it is foamy. Add 1 quesadilla and cook until golden brown on both sides, about 3 minutes per side. Repeat with the remaining quesadillas, adding more butter as necessary.
Cut each quesadilla into 6 pieces. Top with Mexican Cream and Pico de Gallo.
Combine all of the ingredients. Refrigerate until needed.
Over a hot grill, roast the tomatoes until blackened. Transfer to a bowl and add the chipotles. Cover and allow to cool for 15 minutes. Add the basil, garlic and lime juice and marinate for 10 minutes. Transfer to a blender or food processor and pulse until chunky. Transfer to a bowl and add the vinegar, sugar, salt and pepper. Slowly whisk in 1 cup of olive oil. Reserve.