In a large bowl, cream butter. Add eggs, vanilla and sugar.
On low speed, add dry ingredients and mix until combined. Mixture should be mealy.
Divide dough in half, pat it into a disc, and wrap each disc in plastic wrap. Chill for 2 or more hours.
Working with one disc at a time, roll dough to 1/4 inch thick. Cut out large cookies and bake for 12-16 minutes until they are set but not golden brown yet.
Let cool completely before decorating.
Beat vanilla, salt, powdered sugar and 4 tbsp of heavy cream until frothy. Continue beating until the mixture is shiny. Continue beating until mixture holds stiff peaks, about 5-7 minutes. If mixture is too stiff, add cream. If too loose, add powdered sugar.
Add food coloring and transfer to pastry bags. Decorate eggs with icing.