With an electric mixer, combine powdered sugar, butter, milk, and vanilla until smooth.
Add food dye, if desired and mix until color is even.
Add strawberry jam and mix for about 2 minutes to incorporate well. Keep icing at room temperature until cookies are ready to ice.
For Cookies:
Cream the butter and sugar together until light and fluffy. Add the egg and vanilla and beat until well combined.
Add the remaining ingredients and mix until well combined and there is no white flour visible. You may need to scrape down the sides during this step.
Form dough into a disc and chill for at least 2 hours or until well firm.
Roll on a well floured surface to 1/4" thickness and cut into desired shapes. Place on a parchment lined baking sheet and bake for 10-12 minutes, or until browned around the edges.
If desired, cool on a baking rack completely before icing.