Preheat the oven to 450F convection. Line a baking sheet with parchment paper.
To the baking sheet, add the carrots, olive oil, and ½ teaspoon salt. Toss to coat well. Make spaces on the baking sheet to add the chicken and drizzle with a bit more oil, sprinkle with the remaining ½ teaspoon salt, and use your hands to rub it all around. Bake for 25 minutes or until chicken and carrots are beginning to brown or char in spots.
In a large bowl, whisk together the dressing ingredients until smooth. If your peanut butter is cold and it’s not mixing in well, pop it in the microwave for 10-15 seconds and keep whisking. The dressing should be pretty thin, so add water or more vinegar, if needed.
Transfer the chicken to a large cutting board and roughly chop it. To the cutting board, add the cucumber, cabbage, and scallions. Use your knife to chop everything up until it is all roughly in uniform bite-sized pieces (or you can chop them all separately, see note). Transfer to the large bowl along with the warm carrots and toss to coat in the dressing. Taste and add more salt, soy sauce, or vinegar, if needed.
Serve topped with chopped nuts, a squeeze of lime, and sriracha or chili oil (if using).