Preheat the grill to medium-high. Toss the chicken and carrots with the olive oil and salt.
Grill the chicken for 5–7 minutes per side, or until nicely browned and cooked through. Grill the carrots for 4–6 minutes per side, or until lightly charred and tender. The exact timing will depend on their thickness.
Transfer the chicken and carrots to a cutting board and let rest for 5–10 minutes, then chop into bite-sized pieces.
Meanwhile, make the dressing. In a large bowl, whisk together the peanut butter, lime juice, soy sauce, maple syrup, and optional add-ins until combined. Whisk in ⅓ cup warm water until smooth. Add more water, 1 tablespoon at a time, until the dressing is creamy but pourable. Taste and add more lime juice as needed.
Add the chopped chicken and carrots, cabbage, cucumbers, scallions, and herbs to the bowl. Toss until everything is evenly coated. Taste and add more lime or soy sauce, if needed.
Divide among bowls and top with chopped peanuts, spicy topping (if using), and a squeeze of lime.