In a large mixing bowl beat the butter, sugar, salt, 1 whole egg and 1 egg yolk on medium speed until light and fluffy. Scrape down the sides and bottom of the bowl and beat again until blended. Add the vanilla and blend on low until incorporated.
Add the flour and mix on low just until combined. Using a rubber spatula, press the dough together. Divide the dough in half and form each into an 8-inch round disk. Wrap the disks in plastic wrap and refrigerate 1 hour.
Preheat oven to 350°F. Line two baking sheets with parchment paper.
On a large, well floured work surface, roll out one disk of dough to ⅙-inch thick. Using the larger cookie cutter, cut out as many cookies as possible. Cut out the center of each cookie using the smaller ¾-inch cutter. Carefully transfer the cookies to the prepared baking sheets, spacing them 1-inch apart. Refrigerate the baking sheet, with unbaked cookies, for 15 minutes.
Gather the dough scraps and form into a dish. Refrigerate while baking the rolled out cookies.
In a small bowl, whisk the remaining egg white with a pinch of salt and 1 teaspoon water, to make an egg wash.
Remove the baking sheet from the refrigerator and using a pastry brush, gently brush each cookie with the egg wash. Allow the egg wash to dry a minute or two then sprinkle each cookie generously with coarse sugar.
Bake the cookies for 12 to 15 minutes or until the edges start to turn golden brown. Cool the cookies on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely. Repeat with the remaining dough by rolling, shaping, baking and cooling.
Cookies may be stored at room temperature for up to a week, or frozen until needed.