Preheat oven to 375 degrees. Brush a large rimmed baking sheet with oil; set aside.
In a pie plate or shallow bowl, combine cornflake crumbs, cayenne, and 1 tablespoon salt. In a large bowl, toss potatoes with oil to coat. One or two at a time, dip potatoes in crumbs, turning to coat all sides; transfer to prepared baking sheet. Bake potatoes, turning once, until crispy on the outside and tender on the inside, 40 to 50 minutes. Remove from oven, and season with more salt.
Meanwhile, in a medium bowl, combine beef, cheddar, 1 teaspoon salt, and 1/2 teaspoon pepper. With a fork, mix very gently to combine. Gently shape meat into 4 patties, each 4 inches in diameter and 1 inch thick; season with salt and pepper.
Fifteen minutes before potatoes are finished baking, heat a large nonstick skillet over medium-high. Place burgers in skillet, and cook until browned on both sides and done to taste (4 to 6 minutes per side for medium-rare). Serve on hamburger buns with crispy oven fries, Burger Sauce, and desired garnishes.