To make the cookies: In a mixing bowl, combine the cake mix, eggs, and oil with a wooden spoon to make a dough. Form into 24 (1-1/2-inch) balls and put on cookie sheets. Using the palm of your hand, flatten each cookie a bit.
Bake at 350F for 7-8 minutes until they just start to get a little firm around the edges. Transfer to a rack to cool completely.
To make the white chocolate glaze: In a medium mixing bowl (glass is best), microwave the white chocolate morsels and shortening 45 seconds. Whisk until smooth (morsels will continue to melt during this). Whisk in the powdered sugar and milk until smooth.
Take a cooled cookie and dip the top into the white chocolate glaze. Let the excess drip off, then put back of the rack. Sprinkle glazed cookie tops with the blue sugar sprinkles. Allow to sit for the glaze to harden. Yield: 24 cookies.
Over time, the food coloring from the cake mix will bleed into the glaze. So I don't recommend making these further in advance than the day of when you want to serve them.