This looks like a depression-era hot milk cake with mocha frosting. That’s also why you add the eggs later, unless scrambled eggs appeals to you.
From a box sold in Canby, Oregon.
Daisy Cream Cake
1/2 c. shortening
2 c. flour
1-1/4 c. sugar
3 tsp. baking powder
1 tsp. salt
3/4 c. milk
1 tsp. vanilla
2 eggs
Add together everything except egg. Beat 20 minutes, then add egg.
[Frosting]
1/4 c. shortening
3-1/2 c. powdered sugar
1/8 tsp. salt
2 Tablespoons cocoa
Add an egg and 1/8 c. strong black coffee
Beat till fluffy.
Yesterdish suggestion: Traditionally, you would heat the milk and the shortening together, then add it to the dry ingredients, then beat for 20 minutes before adding the eggs. Assuming two 9″ layers, bake at 350 degrees for about 35 minutes.