Make the dressing-marinade. In a bowl, whisk fish sauce, lime juice, soy sauce, rice vinegar, brown sugar, chili-garlic sauce, garlic, ginger, and neutral oil.
Taste and adjust: add more lime for brightness, sugar for balance, or chili for heat.
Marinate the chicken. Place chicken in a zip-top bag or shallow dish. Pour in about 1/3 of the dressing. Toss to coat and chill for 20–30 minutes (up to 4 hours).
Reserve the remaining dressing for the salad.
Prep the salad base. In a large bowl, combine cabbage, greens, cucumber, bell pepper, carrot, mint, cilantro, and Thai basil. Keep chilled until ready to assemble.
Prep the mango and toppings. Slice mango into thin strips or small cubes. Chop peanuts or cashews.
Slice green onions and chili. Set aside.
Heat the grill or pan. Preheat a grill to medium-high and oil the grates, or heat a grill pan or cast-iron skillet with a light film of oil.
Grill the chicken. Remove chicken from marinade, letting excess drip off. Grill 4–6 minutes per side (thighs) or 4–5 minutes per side (breasts), until cooked through and nicely charred.
Internal temp should reach 165°F (74°C).
Rest and slice. Let chicken rest for 5 minutes, then slice thinly across the grain.
Toss the salad. Add mango and most of the nuts to the salad bowl. Drizzle with the reserved dressing and toss gently. Aim for light, even coating—don’t drown it.
Assemble and serve. Top the dressed salad with sliced chicken.
Sprinkle with remaining nuts, green onions, and chili. Serve with lime wedges. Enjoy right away.