Preheat oven to 325 °F (160 °C). Line two baking sheets with parchment paper and position a rack in the upper third of the oven.

In a medium bowl, whisk butter until very light and fluffy (~20–30 seconds). Stir in powdered sugar until fully incorporated, followed by cornstarch, vanilla, and salt; mix until smooth. Add flour and stir until dough comes together—it may feel crumbly at first.

Scoop dough by weight (~30 g) or use a small cookie scoop to form balls. Roll each ball in granulated or cane sugar to coat. Arrange on sheets spacing ~2 inches apart, then gently flatten each ball to ~½″ thick using a flat-bottom glass.

Bake for about 12 minutes, until edges are just set and bottoms are barely golden. Let cool on sheets for a few minutes before transferring to wire racks.

While cookies bake, microwave strawberries with sugar in a microwave-safe bowl for 1 minute. Strain through a fine-mesh sieve, pressing solids to extract juice. Microwave juice again until reduced to ~2–2½ Tbsp.

Once cookies are fully cool, whisk reduced syrup with lemon juice, powdered sugar and vanilla until you have a thick but pourable icing. Adjust consistency with more sugar or lemon juice as needed.

Spoon or drizzle icing over each cooled cookie. Let set before serving.