In a large bowl, whisk together the mayonnaise, vinegar and sugar. Add the coleslaw and toss; season with salt and pepper. Refrigerate until ready to serve.
Season the turkey cutlets with salt and pepper. In a large skillet, heat 1 tbsp. EVOO over medium-high heat. Working in batches, using more EVOO as necessary, cook the turkey, turning once, until golden, 3 minutes. Transfer to a plate. Pour the broth into the skillet; simmer for 1 minute. Off the heat, return the turkey and any juices to the pan.
On a foil-lined baking sheet, broil the bread until toasted. Set the tops aside. Top the bread bottoms with the turkey and cheese; broil until melted, 1 to 2 minutes.
Spoon some of the skillet juices onto the bread tops; place on the sandwiches. Strain the remaining juices into a bowl for dipping. Serve with the coleslaw.