Toss the potato wedges with olive oil, dried basil, paprika, cayenne, pepper, and salt in a large ziplock bag.
Use tongs to place the potato wedges in a single layer with skin side facing down on a foil-lined baking tray.
Bake the potato wedges at 400 F for 45 minutes.
Take the baking tray out of the oven. Rotate the potato wedges so that they are laying on their sides, with little or no gap between adjacent wedges. Sprinkle cheddar cheese over them. Return them to the oven and bake until the cheddar cheese melts, about a few minutes.