Combine butter, cream cheese, and sugar in a large bowl, and cream with an electric mixer for several minutes until light and fluffy. Beat in egg and vanilla.
Whisk flour, baking powder, and salt in a separate bowl. Slowly add the dry ingredients to the butter mixture until just combined.
Divide the dough into 2 balls. Place each between two pieces of parchment (to prevent sticking) and roll out to about 1/8 inch thick. Transfer dough to cookie sheets on the parchment; refrigerate 30 minutes.
Heat oven to 350°F. Use cookie cutters to cut shapes, pulling away the scraps to avoid moving and distorting the cookies. If the dough gets too soft to work with, refrigerate it for 15 to 30 minutes. Reroll scraps between sheets of parchment for more cookies.
Bake 7 to 10 minutes, or until the cookies just start to color at the bottom edges. Let cool for 5 minutes on the sheet, then remove to a wire rack to cool completely before decorating.