Make the Gyro Meat:
In a small bowl combine paprika, salt, pepper, and oregano.
Slice the meat against the grain as thin as possible and pound with a meat mallet to less than 1/4-inch thick. In a flat-bottomed container, arrange one layer of slices, sprinkle liberally with seasoning mixture, then with a little vinegar. Spread the vinegar over the meat so that all pieces are moistened. Add layers, spices, and vinegar until all the meat has been marinated. Cover and refrigerate for 1/2 hour to 2 hours.
Make the tzatziki and french fries.
Remove meat from the refrigerator and slice into strips about 1/2 to 3/4 of an inch wide and 2 to 2 1/2 inches long. Dry fry (without any oil) in a non-stick frying pan until browned and slightly crisp.