Make the chicken marinade: Add all of the chicken marinade ingredients to the bowl of a food process (or a blender would work). Process in the food processor to combine well. Remove the marinade to a medium bowl.
Butterfly the chicken by cutting in half from the side, cutting almost all the way through, but not all the way through. Open the chicken up like a book and press down to flatten. Add the chicken to the marinade and let stand at least 15 minutes or cover and refrigerator for up to 3 hours.
Prepare the peanut sauce: Rinse out the food processor or blender. Add the peanuts and pulse the peanuts briefly until finely ground. Add the garlic, soy sauce, sesame oil, brown sugar, fish sauce, lime juice, cayenne and coconut milk and process until well puréed. Transfer the peanut sauce to a small bowl. Taste and add a bit of salt at this point if needed. Set aside if using immediately or cover and refrigerate for up to 2 days. The peanut sauce will thicken in the refrigerator. To thin, drizzle in a little hot water until the sauce is thinned enough to run off the end of a spoon.
Preheat the BBQ or place a grill pan into the oven and preheat with the oven to 450F (non-convection) oven.
Place the marinated chicken onto the BBQ or the hot grill pan in the oven and cook, flipping once, until the chicken reaches 165F internal temperature, about 12-15 minutes. Remove the chicken to a cutting board and let it rest a few minutes before slicing into strips.
Make the cucumber salad: While the chicken is cooking, add all the cucumber salad ingredients to a bowl. Stir together to combine and set aside.
Assemble the salad: Place a layer of lettuce onto the bottom of a serving platter. Spoon the cucumber salad on top to one side of the plate. Place the sliced chicken on the other side of the plate. Drizzle with peanut sauce.
Serve immediately with additional peanut sauce on the side and some lime wedges for drizzling.
Notes
Note 1: Unsalted or lightly salted peanuts are recommended to avoid an overly-salted finished dish. (Peanuts in the shell are dry roasted and are often unsalted.) If you only have salted peanuts, using low-sodium soy sauce and reducing the Fish sauce slightly can reduce the overall saltiness. If you don't have peanuts on hand, substitute natural peanut butter, preferably a low-salt peanut butter. In a pinch you can use regular peanut butter. As it is both salted and sweetened, adjusting the additional salty elements can help bring it more back into balance.Note 2: Leftover coconut milk can be frozen for up to 3 months!Note 3: If you don't have sweet chili sauce on hand, mix together a bit of honey, red pepper flakes and a splash of hot sauce instead.