Preheat oven to 400°.
Cut the onion crosswise into 3/4-inch-thick slices, and separate into rings. Use 16 of the largest rings; reserve remaining onion for another use. Lightly spoon flour into a dry measuring cup; level with a knife. Combine flour, salt, paprika, and pepper in a medium bowl. Stir in beer and egg white (batter will be thick). Heat 1 1/2 teaspoons oil in a large nonstick skillet over medium-high heat. Dip 5 onion rings in batter, letting excess drip off. Add onion rings to pan; cook for 2 minutes on each side or until golden. Place the onion rings on a jelly-roll pan. Repeat procedure of dipping onion rings in batter and cooking in remaining oil, ending with 6 rings. Coat the onion rings with cooking spray. Bake at 400° for 10 minutes or until crisp. Serve rings with ketchup.