In a large, deep pot, heat 4 inches of oil until it reaches 350 degrees F. Bring milk to a boil in a large saucepan with butter, salt, cayenne, and white pepper. Slowly add flour, whisking constantly and vigorously until the mixture begins to pull away from the sides of the pan, about 5 minutes.
Cover an 11 by 17-inch baking sheet with plastic wrap. Pour mixture onto 3/4 of the prepared baking sheet, spreading evenly with an offset spatula. Cover with plastic wrap, and transfer to refrigerator to chill until firm, about 1 hour.
Remove plastic wrap, and cut crosswise into 3-inch strips. Cut strips crosswise into 1/2-inch fries. Transfer fries to heated oil, and fry in batches until golden, about 2 minutes. Remove from oil, and season with salt and cayenne pepper. Repeat with remaining fries. Serve immediately.
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