Preheat the broiler. In a small bowl, combine the cheese, pimientos and mayonnaise; season with salt and pepper.
In a large bowl, toss the chicken with the cabbage and radishes. Add the olive oil and vinegar; season with salt and pepper. Divide the chicken slaw among 4 shallow bowls.
Spread some pimiento-cheese mixture on each slice of cornbread. Transfer to a baking pan and broil until the cheese is browned and bubbling. Serve alongside the chicken slaw.