In a large bowl, with the mixer at medium speed, mix the butter and sugar until light and fluffy, about 3-4 minutes. Add vanilla and egg, mix until well combined. Scrape the sides of the bowl as needed.
In a medium bowl, whisk together flour and salt. Add to butter-sugar mixture. Mix at slow speed until just combined. Wrap the dough in plastic wrap and chill for 30 minutes.
Preheat oven to 350°F. Line three baking sheets with parchment paper.
Remove the dough from the refrigerator and roll the dough about ¼-inch thick. Use a cookie cutter to cut out shapes. Place on prepared baking sheets and bake for 12-14 minutes until golden brown.
Allow the cookies to rest on the baking sheet for 5 minutes and then transfer to a cooling rack. Allow the cookies to cool completely before icing.
Prepare the icing. Combine all ingredients together in mixing bowl, except the food coloring.
Beat on low speed until combined, then increase speed to medium high and beat for 5 – 8 minutes, adding 2-3 tablespoons more warm water, as needed. Icing should be stiff enough to hold a peak when tested.
Add food coloring of your choice, adjusting the amount depending on the intensity of the color you desire. Use this thick icing to outline cookies.