Prepare the gravy: In a small bowl, dissolve the cornstarch in the water and set aside.
In a large saucepan, bring the chicken and beef broth to a simmer.
Whisk in about half the cornstarch mixture and simmer for a minute. If you'd like your gravy thicker, add a more of the cornstarch mixture, in small increments, as needed, to thicken. Season with pepper. Taste and add additional salt, if necessary, to taste. Make ahead and re-warm or keep warm until your fries are ready.
Cut the potatoes into 1/2-inch thick sticks. Place into a large bowl and cover completely with cold water for 15 minutes. Drain well and pat dry with a towel before frying. Heat the oil in a heavy bottom pot to 300° F.
Add your fries to the 300°F oil and cook for 5-8 minutes, just until potatoes are starting to cook but are not yet browned. Remove potatoes from oil and scatter on a paper towel lined baking sheet. Increase oil temperature to 375°F. Once the oil is heated to that temperature, return the potatoes to the oil and cook until golden brown. Remove to a paper towel-lined bowl.
Add the fries to a baking sheet. Season lightly with salt while still warm. Spread the cheese over top. Place underneath the broiler until the cheese is melted. Ladle gravy over top and serve.