In a medium bowl, combine flour, salt and baking powder. Set aside.

In large mixing bowl, beat butter with an electric mixer on medium to high speed for 30 seconds. Add powdered sugar and vanilla. Beat until smooth, scraping sides of bowl as needed. Beat in as much of the flour mixture as you can with the mixture; using a wooden spoon, stir in any remaining flour mixture.

Divide dough in half. On a large piece of waxed paper, shape one portion of dough into a 16-inch log that is about 1 inch in diameter. Cut log in half crosswise. Sprinkle half of the Demerara sugar on the sheet of waxed paper; roll each log in sugar to coat evenly. Wrap each log individually in plastic wrap. Repeat with remaining portion of dough and remaining Demerara sugar. Chill logs for 1 to 24 hours or until firm enough to slice.

Reshape logs if needed. With a thin sharp knife, cut each log into 16, 1/2-inch thick slices. (Rotate logs while cutting them to prevent flattening one side.) Place slices about 1 inch apart on parchment paper-lined, foil-lined, or ungreased cookie sheets. Bake in a 350 degree F oven for 11 to 13 minutes or until tops are firm and edges are just starting to brown. Transfer cookies to wire racks to cool. Store in an airtight container at room temperature for up to 3 days or in the freezer up to 3 months. Makes 64 cookies.

Prepare as above, except add 1/2 cup unsweetened cocoa powder with the flour mixture and increase powdered sugar to 3/4 cup. Bake cookies as directed above, or until tops are firm (edges will not start to brown).