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.
Unless for some unknown reason you have an open can of beer in your refrigerator, try this simple way to flatten the suds you need for this batter. Measure 1/2 cup beer in a small bowl, and stir with a fork. You're left with about 1/3 cup flat beer.