Preheat oven to 350º. Oil a 8"x8", 9"x9, 7",8" or 9" round cake pan or a pie pan.
In a large mixer bowl, mix together eggs and sugar on medium-low, or whisk by hand.
Add butter in a little bit at a time, as to not cook eggs; add vanilla and milk; mix again on medium-low.
Add flour; beat until smooth.
Reserve about 2/3 c. of batter; spread just enough batter/dough to cover the bottom of the prepared baking pan.
Drop 1/2 T. of jam or pie filling over the batter; spread as much as possible.
Drop the remaining batter by heaping tablespoonfuls onto the top of the filling and spread a bit for some of the spoonfuls to meet, leaving a bit of apple showing.
Bake in 350º oven for 45 minutes or until toothpick clean in the center. Cover with foil if it becomes too brown.
White the cake is baking, add powdered sugar to a cup; add melted butter and stir until sugar is melted and well mixed.
Add milk and vanilla and stir well; baste over warm coffee cake.
Serve warm. Store leftovers covered.
Refrigerate if not using right away within a couple of days.
Options: fillings may include jam, peach, apple or cherry pie filling, cinnamon and sugar, custard, cranberry sauce, etc.