In a bowl, beat the eggs until pale yellow. Add the sugar and vanilla and beat until thick and light-colored. Add the oil and milk, beating well to combine.
Sift the flour and baking powder together and stir into the egg mixture.
The dough will be soft; wrap it in wax paper and refrigerate for at least one hour.
Preheat the oven to 350 degrees. Lightly grease three cookie sheets.
Place the dough on a well-floured surface. Knead dough for about 5 minutes, or until smooth and no longer sticky.
Break off egg-size pieces of dough and roll or coil into balls. Place the dough balls one-inch apart on the cookie sheets.
Bake for 20 minutes, or until lightly colored and firm to the touch.
Transfer cookies to wire racks to cool slightly before frosting.
For frosting, combine the sugar, milk and vanilla and stir until smooth.
While the cookies are still warm, dip the top of each cookie in the frosting. Place the cookies on racks to let the frosting dry before storing. Sprinkle with nonpareils, if desired.
Cookies will keep in an airtight container for up to one week or frozen for up to three months.