In large bowl, combine brown sugar, butter, vanilla and egg; beat until light and fluffy. Add flour and 3/4 teaspoon apple pie spice; mix well. Divide dough in half; shape each half into flattened disk. Wrap each in plastic wrap; refrigerate about 1 hour for easier handling.
Heat oven to 350°F. On lightly floured surface, roll half of dough at a time to 12x6-inch rectangle. Keep remaining half of dough refrigerated. Cut dough rectangle lengthwise in half; cut each half into 1-inch-wide sticks. Place sticks 1/2 inch apart on ungreased cookie sheets.
Bake at 350°F. for 12 to 15 minutes or until edges are light golden brown. Immediately remove from cookie sheets. Cool 10 minutes or until completely cooled.
Meanwhile, in small bowl, blend all glaze ingredients until smooth, adding enough apple juice for desired glaze consistency. Spread glaze on tops of cookies. Let stand until glaze is set before storing.