Preheat oven to 350 degrees F. Line a baking sheet with parchment paper or a silicone baking mat; set aside.
In the bowl of an electric mixer fitted with the paddle attachment, beat butter and sugar on medium-high until light and fluffy, about 2-3 minutes. Beat in vanilla until well combined. Beat in eggs, one at a time, until well combined. Gradually add flour and salt to sugar mixture at low speed, beating just until incorporated.
Roll the dough out between parchment paper. Cover and place in the refrigerator for at least 1 hour.
Using a Christmas tree cookie cutter, cut out cookie shapes and place onto prepared baking sheet.
Place into oven and bake until edges are golden brown, about 8-12 minutes depending on thickness of cookies.
Remove from oven and cool on a wire rack.
In the bowl of an electric mixer fitted with the paddle attachment, beat butter on medium speed until smooth, about 3 minutes. With the mixer on low speed, gradually add powdered sugar, beating just until incorporated. Add vanilla and milk. Increase speed to medium high and beat until frosting is light and fully, about 1-2 minutes.