Beat eggs slightly, then add oil, baking powder, flour and sugar. Mix well and refrigerate overnight or for several hours.
When your dough is cool. Roll it in your hands to make walnut sized balls or smaller. Place them on a cookie sheet and bake at 350 for 15-18 minutes.
While these are baking, make your vanilla pudding according to the package directions.
When they have cooled, use a small melon baller to scoop out the centers of two balls and fill each in with a spoon of pudding. Put the two cookies together. Continue doing this until all of your pudding filled balls have been made.
In a bowl mix 1 cup of Sweet Vermouth and 1/4 tsp of orange food coloring. Dip each cookie into the Vermouth mixture and then set on a rack to drain, but do not let it dry. After each cookie sits on the rack for a minute or so, roll it in a bowl of granulated sugar, and set back on the rack to dry.
Now that you have your peaches, you are going to make a quick butter cream to pipe "leaves" onto the top of each one. Mix together your confectioner's sugar, butter, milk and green food coloring. Use a decorator bag and a leaf tip to pip one or two small green leaves at the top of each cookie.
After these have "set" I put them in fancy little paper cups for serving! Yumm