Peel and slice onions. Heat 1 teaspoon butter in a large skillet. Add sliced onion, saute, stirring until just golden and tender. Preheat oven to 300 degrees, spoon onion onto a heatproof platter; keep warm. Combine 1/2 cup flour and bread crumbs in plastic bag. Beat egg in a small bowl; add salt, paprika, and basil.
Wipe steaks; dip each in egg then in crumb mixture, to coat completely. Melt 1 tablespoon butter in the skillet. Saute steaks until golden brown on one side - 3 minutes. Turn; saute on remaining side until of desired doneness - 3-5 minutes. Arrange on onion.