For cookies: In the bowl of a stand mixer fitted with the paddle attachment, beat butter, granulated sugar, and lemon zest at medium speed until creamy, 3 to 4 minutes, stopping to scrape sides of bowl. Add salt and vanilla extract, and beat at medium-low speed until combined.

In a medium bowl, whisk together flour and finely ground strawberries. Add flour mixture to butter mixture, and beat at low speed until combined and dough starts to come together. (Dough will be crumbly.) Knead dough by hand in bowl until it comes together into a ball. Turn out dough onto a piece of plastic wrap. Using your hands, shape and roll dough into a 2½-inch-thick log (about 7 inches long). Wrap in plastic wrap, and refrigerate until firm, about 30 minutes.

Preheat oven to 300°F (150°C). Line baking sheets with parchment paper.

Using a sharp knife, trim ¼ inch from ends of log. Cut log into 15 slices (about 3/8 inch each). Place 2 inches apart on prepared pans. Freeze until firm, about 10 minutes.

Bake until tops look dry and are set, about 14 minutes. While cookies are still hot, gently swirl a 4- to 5-inch round cutter in a circular motion around each cookie to create an even round. Let cool on pans for 10 minutes. Remove from pans, and let cool completely on a wire rack.

For glaze: In another medium bowl, whisk together confectioners’ sugar, 5 teaspoons (25 grams) water, vanilla powder, and salt until smooth; drizzle onto cooled cookies as desired. Immediately garnish with ground strawberries, if desired. Let stand until glaze is set. Store in an airtight container for up to 3 days.