Beat butter and sugar at medium speed with an electric mixer 2 minutes or until creamy. Add eggs, 1 at a time, beating until smooth, stopping to scrape down sides as needed. Stir in vanilla. Gradually add flour to butter mixture, beating well at low speed until blended.
Sprinkle sugar on wax paper. Shape dough into 2 (6 1/2-inch) logs on sugar; roll logs in sugar. Wrap each log in wax paper, and chill 4 hours. (Freeze dough in zip-top plastic freezer bags up to 1 month, if desired. Thaw 10 minutes before slicing.)
Cut each log into 28 slices, and place 1 inch apart on lightly greased or parchment paper-lined baking sheets.
Bake, in batches, at 350° for 10 to 12 minutes or until edges are golden. Let cool on pans 2 minutes. Transfer cookies to wire racks, and let cool completely.
Ginger Sand Dollar: Stir 1/2 cup crystallized candied ginger, finely chopped, into batter. Cut logs into slices, sprinkle each with sugar, and arrange 5 slivered almonds in a star design to resemble sand dollars. Slice and bake as directed. Makes about 5 dozen.
Nantucket Cranberry-White Chocolate Cookies: Stir 1 cup dried cranberries and 1 cup white chocolate morsels into batter. Slice and bake as directed. Drizzle 6 ounces white chocolate, melted, over cooled cookies; sprinkle with nonpareils. Let stand until chocolate is firm. Makes about 5 dozen.