Heat approximately 3 or 4 inches of oil in a deep fat fryer to 370 degrees F.
While the oil is heating, peel potatoes and cut into strips using a French fry cutter.
Place potato strips in ice water to remove excess starch.
Completely dry potato strips using paper towels.
Divide potatoes into two batches.
Fry each batch in hot oil until golden brown; approximately 5 to 6 minutes/batch.
While the French fries are cooking, make the Ultimate Dipping Sauce by pouring the mayonnaise on one side of a small serving bowl and the BBQ sauce on the other side of the serving bowl. DO NOT mix the mayonnaise and the BBQ sauce; mixing it with the French fries is part of the experience.
Remove cooked fries with a slotted spoon and place on paper towel to remove excess oil.
Remove paper towel from plate.
Sprinkle grated cheddar cheese on piping hot French fries.
Serve immediately, but allow the cheese to melt before serving. You may want to place under the broiler for a minute - just until the cheese melts; DO NOT let the cheese bubble and crust over!
Dip the cheese fries into the Ultimate Dipping Sauce making sure you get a little bit of mayonnaise and a little bit of the hickory flavoured BBQ sauce.