In a large bowl, cream together the sugar and butter until smooth. Beat in the eggs and egg yolks, one at a time, mixing well after each. Mix in vanilla extract. Combine flour, baking powder, and salt; stir into the sugar mixture. Cover dough and chill for at least 1 hour.
Preheat the oven to 375 degrees F (190 degrees C). Grease cookie sheets. On a floured surface, roll the dough out to 1/2 inch thickness and cut into desired shapes using cookie cutters. Place 2 inches apart on the prepared baking sheets.
Bake for 8 to 10 minutes in the preheated oven. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack.
To make frosting, mix together the confectioners' sugar, oil and 1 teaspoon vanilla until smooth. gradually add enough hot water to achieve a spreadable consistency. Spread over warm cookies.