In a food processor fitted with the shredding disk, grate the broccoli florets and stems and the carrots.
In a large bowl, whisk the buttermilk, mayonnaise and lemon juice. Add the broccoli and carrots; toss to coat and season with salt and pepper.
Divide the chicken and the broccoli slaw among plates. Sprinkle the slaw with the capers and lemon zest.