Chicken Preparation: In a mixing bowl, mix together coconut milk, garlic, cilantro, ginger, coconut aminos, chile pepper, lemongrass, and turmeric until well-combined.
Using a sharp knife, cut chicken breasts into strips about ½-inch thick to form mini-fillets. Transfer cut chicken to bowl of marinade and toss until chicken is fully coated in sauce. Cover bowl with foil or plastic wrap and refrigerate to marinate for at least 1 hour, preferably overnight.
Remove marinated chicken from refrigerator. In a large pan or griddle over medium heat, heat olive oil. Using tongs, place chicken in an even layer in pan. Cook chicken until internal temperature of 165 degrees is reached, about 3-4 minutes per side. Alternatively, cook chicken in preheated oven at 375 degrees until cooked throughout, about 20-30 minutes. Set aside.
Dressing: In a small mixing bowl, whisk together dressing ingredients. Set aside.
Salad: Using a julienne peeler or spiralizer, carefully cut cucumbers into cucumber noodles. (If using a julienne peeler, discard the center of the cucumbers.)
Final Steps: In a large mixing bowl, using spoons or tongs, toss together arugula, cucumber noodles, green onions, and dressing until thoroughly incorporated. Serve dressed greens mixture in individual bowls and top with chicken, chopped cilantro, and finely sliced fresh chile pepper.