1) To make the white dough: Beat together the sugar, butter, salt, and vanilla.
2) Add the flour, stirring to make a cohesive dough. Divide the dough in half, flatten each half into a rectangle that's about 5" x 7", wrap the pieces in plastic, and refrigerate for 2 to 4 hours.
3) To make the red dough: Beat together the sugar, butter, salt, vanilla, and color.
4) Add the flour, stirring to make a cohesive dough. Divide the dough in half, flatten each half into a rectangle that's about 5" x 7", wrap the pieces in plastic, and refrigerate for 2 to 4 hours.
5) To make the glaze: Whisk together the egg white and water until slightly foamy. Refrigerate until ready to use.
6) To shape the cookies: Place parchment or waxed paper on your work surface and dust it lightly with flour.
7) Lay one square of white dough on the parchment and roll it into a 12" x 6" rectangle. Brush with the glaze, and set it aside.
8) Using another piece of parchment or waxed paper, roll one square of red dough slightly smaller than the first; it should be about 11 1/2" x 5 1/2".
9) Place the smaller piece of red dough on top of the larger piece of white dough, using the parchment to transport.
10) Starting with a long edge, use the parchment to help you gently roll the stacked dough into a tight log, with no gaps.
11) Press the edge to seal, then wrap the log in plastic wrap or parchment, and freeze for at least 1 hour.
12) Repeat with the second red and white pieces of dough.
13) Preheat the oven to 350°F. Lightly grease (or line with parchment) two baking sheets.
14) Remove the log from the freezer, unwrap it, and place it on a clean work surface.
15) Use a sharp knife to gently cut the log into 1/4" to 3/8" slices. Transfer the cookies to the prepared baking sheets.
16) Bake the cookies for 12 to 14 minutes, or until they feel firm. Remove them from the oven and transfer to a rack to cool.
17) When cool, store airtight at room temperature for up to a week; freeze for longer storage.
Yield: about 4 dozen cookies.