Slice each chicken breast on the diagonal against the grain into four or five 1/2-inch-thick medallions. In a medium bowl, whisk together 2 tbsp. EVOO, 2 tbsp. mustard, 1 1/2 tbsp. vinegar, 1 tbsp. water, the thyme and 1/4 tsp. each salt and pepper. Add the chicken pieces, turning to coat. Cover and refrigerate for at least 1 hour and up to 24 hours.
At least 30 minutes before cooking the chicken, make the slaw. In a large bowl, whisk together the remaining 2 tbsp. EVOO, 1 tbsp. mustard, 1 tbsp. vinegar and the brown sugar until blended. Add the cabbage, bell peppers and scallions and toss well. Season to taste with salt and pepper.
Preheat a grill to medium. Lightly spread the cut sides of the rolls with butter. Grill, buttered side down, until lightly toasted, 1 to 2 minutes. Transfer to plates or a platter.
Add the chicken to the grill, cover and cook, turning once, until lightly charred on both sides and just cooked through, about 5 minutes. Transfer the chicken pieces to the roll bottoms, top with the slaw and cover with the roll tops.