To prepare cookies, weigh or lightly spoon flour into dry measuring cups; level with a knife. Add flour and next 5 ingredients (through butter) to the bowl of a stand mixer. Beat at low speed until crumbly. Add egg white and water; beat until blended.
Divide dough in half. Roll each half of dough into an 8 1⁄2 x 1 1⁄2–inch log. Wrap in plastic wrap; chill 2 hours.
Preheat oven to 350°.
Cut each log into 48 (1⁄8-inch-thick) slices. Place on baking sheets lined with parchment paper.
Bake at 350° for 10 to 12 minutes or until set. Cool on pans 2 minutes; transfer cookies to wire racks, and cool completely.
To prepare filling, beat shortening and 1⁄3 cup butter with a mixer at medium speed until smooth. Gradually add powdered sugar, beating at low speed until blended. Beat in vanilla.
Spread about 1 1⁄2 teaspoons filling onto flat side of each of 48 cookies; top with remaining 48 cookies, flat sides down.
Note: You can freeze wrapped logs of dough for up to 1 month. Freeze prepared sandwich cookies in a heavy-duty zip-top plastic bag for up to 3 months. The cookies will keep in an airtight container at room temperature for up to 1 week.