Beat butter at medium speed with an electric mixer 2 minutes or until creamy. Gradually add sugar, beating well. Add egg and vanilla, beating well. Gradually add flour and salt, beating until blended. Divide dough in half; cover and chill 1 hour.
Roll each portion of dough to 1/4-inch thickness on a lightly floured surface. Cut with cookie cutters. Place on lightly greased baking sheets.
Bake at 350° for 8 to 10 minutes or until edges of cookies are lightly browned. Cool cookies 1 minute on baking sheets; remove to wire racks to cool completely.
This is the best sugar cookie recipe I have ever made (even better than my Grandma's, and that is saying A LOT!). I like a soft sugar cookie that is nice and buttery good, and this one is it! Great with icing and easy to cut out with cookie cutters. Highly recommend!