Pflaumekuchen, my family's german plum cake

Ingredients

• 15 Italian Prune Plums, halved and stone removed, but not peeled
• 1 stick butter, plus either 1/4 cup shortening or another 1/4 cup butter, softened
• 1 cup sugar
• 1 egg
• 1 teaspoon baking powder
• 1 teaspoon vanilla
• pinch salt
• 1 cup flour
• 1 1/2 tablespoons milk