If using wooden skewers, soak them in water while you prepare the chicken. This helps reduce scorching on the grill.
In a medium bowl, stir together the sweet soy sauce, oil, 2 tablespoons (30 ml) lime juice, garlic, and ginger. Add the chicken and toss until evenly coated. Let it stand while you make the sauce, or refrigerate it for up to 30 minutes.
For the peanut sauce, combine the peanut butter, coconut milk, soy sauce, brown sugar, 1 tablespoon (15 ml) lime juice, and optional chili sauce in a small saucepan. Warm over low heat, stirring until smooth. Stir in warm water a little at a time until the sauce is thick but spoonable. Keep warm on very low heat or set aside.
Heat a clean grill or grill pan over medium-high heat. Thread the chicken onto the skewers, leaving a small space between pieces so the heat can circulate.
Lightly oil the grill grates or grill pan. Grill the skewers, turning every few minutes, until browned in spots and the chicken reaches 165°F (74°C) in the thickest piece, about 8 to 10 minutes total.
Transfer the cooked skewers to a clean platter and let them rest briefly. Sprinkle with cilantro if desired and serve with the peanut sauce and lime wedges.
Serve the chicken satay with steamed rice, cucumber slices, and a simple vegetable side. Keep any sauce that touched raw chicken separate and discard it.