Naty, here are three recipes that I have used with good results. The first two are for unbaked cookies, the other is for cookies that have been baked already. The last one would be my choice for white icing.

Corn Syrup Paint:

4 tsp corn syrup
1 tsp water
4 different paste colors or liquid food coloring

Put 1 tsp of corn syrup and 1/4 tsp water into each of 4 custard cups. Tint each with one of the food colors. Stir well. Paint unbaked sugar cookies with a small paintbrush. Bake as directed in the cookie recipe.
My notes: the paint turns out tasty and bright, but not shiny.

Egg Yolk Tempra:

1 egg yolk*
1 tsp water
5 drops food coloring

Mix egg yolk and water. Divide into custard cups for however many colors you want. Add food coloring. Paint on unbaked cookies and bake according to recipe directions. Cookies should come out with a shiny colored finish.

Sugar Cookie Icing:

1 cup powdered sugar
2 tsp milk
2 tsp light corn syrup
1/4 tsp almond extract
food coloring

Stir together the powdered sugar and the milk until smooth. Beat in the corn syrup and almond extract until the icing is smooth and glossy. If the icing is too thick, add more corn syrup by drops. Divide into separate bowls and add food colorings. Dip baked cookies, or paint with a small paintbrush. Allow to dry completely.
Note: Dries hard, bright, and shiny. Awesome on gingerbread cookies.

Editor's note:
*Cooking Egg Yolks for Use in Recipes
Because egg yolks are a fine growth medium for bacteria, cook them for use in mayonnaise, Hollandaise sauce, Caesar salad dressing, chilled souffles, chiffons, mousses and other recipes calling for raw egg yolks. The following method can be used with any number of yolks.

In a heavy saucepan, stir together the egg yolks and liquid from the recipe (at least 2 tablespoons liquid per yolk). Cook over very low heat, stirring constantly, until the yolk mixture coats a metal spoon with a thin film, bubbles at the edges or reaches 160° F. Immediately place the saucepan in ice water and stir until the yolk mixture is cool. Proceed with the recipe.