Here’s a version from the October 6, 1871 edition of the Rochester (Indiana) Union Spy:
— One Egg Cake. — This makes a very good cake, and not expensive: One egg, one cup of sugar, one and a half cups of flour, six tablespoons of melted butter. If you use baking powder, take a heaping teaspoonful; if not, take one-half teaspoonful of soda, and one of cream tartar. Add flavoring.
A bit more detail was provided in 1884’s Mrs. Lincoln’s Boston Cook Book by Mary Johnson Bailey Lincoln:
One Egg Cake.
1/2 cup butter, creamed
Mix in the order given, and bake in a shallow pan, in a moderate oven, thirty minutes.
From the box of J.L. from Westborough, Massachusetts.
One Egg Cake
2 cups flour
2 teaspoons baking powder
1/4 teaspoon salt
4 Tablespoons butter
1 cup sugar
1 egg (unbeaten)
3/4 cup milk
Sift flour, salt, and baking powder three times. Cream butter; add sugar gradually. Cream together until light and fluffy. Add egg and beat well. Add flour alternately with milk. Beat after each addition until smooth.