Preheat a grill pan to medium-high. Preheat the oven broiler.
In a small bowl, combine the espresso, sugar, mustard, garlic powder, salt and paprika. Mix well. Pull the beef apart on a clean work surface and sprinkle evenly with salt and pepper. Mix well but lightly and form into patties the same size as the rolls. Sprinkle both sides of the patties with 1 teaspoon of the rub each, and pat down to adhere. Refrigerate until ready to grill.
Cut the radicchio into quarters, leaving the core intact. Put the quarters into a medium bowl and pour 2 tablespoons olive oil evenly over the radicchio. Sprinkle with 1/2 teaspoon salt and 1/4 teaspoon pepper. Toss lightly to coat.
When the grill is barely starting to smoke, grill the radicchio, cut side down, until charred, about 2 minutes. Turn to the opposite cut side and grill for an additional 2 to 3 minutes. Remove to a bowl to cool.
Cut the apple into matchstick-size pieces and add to a large bowl. Thinly slice the red onion to make 1/3 cup, and add to the bowl. Remove the cores from the radicchio and thinly slice then add to the bowl with the apple along with the vinegar, 2 tablespoons olive oil, sugar, 1/2 teaspoon salt and 1/4 teaspoon pepper. Combine well and set aside.
Grill the burgers or until you see a nice brown edge forming. About 6 minutes. Turn and top with cheese. Tent the burgers with foil or a hotel pan and pour 2 tablespoons of cold water on grill to create steam to melt the cheese. Turn the grill off.
Slice the rolls in half and brush with olive oil and season with salt. Broil until brown and crisp. Uncover the burgers and turn the grill back on to high. When burgers start a good sizzle, transfer to the rolls and top with slaw before serving. Enjoy!
A viewer, who may not be a professional cook, provided this recipe. The Food Network Kitchens chefs cannot make representation as to the results.