For the cookies: Beat the butter in a large bowl with a handheld mixer until fluffy. Scrape down the sides of the bowl, add brown sugar, and continue beating until light, 2 to 3 minutes. Add egg and vanilla extract, beating until smooth. Gradually add the 2 cups flour and salt while mixing slowly to make a rough dough. Press dough into a ball by hand. Lightly dust the cookie dough with flour and roll it between 2 pieces of waxed paper into a 1/4-inch-thick disk. Slide the disk in the paper onto a cookie sheet and freeze until firm, about 20 minutes (or refrigerate about 1 hour).
Evenly space the racks in the oven and preheat to 350 degrees F.
Transfer the dough to the work surface and remove the top sheet of paper. Cut into cookies with a 4-inch flower or other decorative cutter and transfer them to a parchment-lined or nonstick baking sheet. Press excess dough together and roll and cut into cookies as well. Bake until cookies' edges are golden, about 20 minutes. Cool on a rack.
For the icing: Whisk the water and egg white powder in a medium bowl until foamy and smooth. Gradually whisk in the confectioners' sugar to make a smooth icing. Spread a layer of icing over the entire surface of the cookies (or just in the center, if desired) with the back of a teaspoon. Place sprinkles, chocolate chips, or candies in the center of the cookies. Let rest until the icing sets, about 30 minutes.
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