Begin by melting ½ cup of unsalted butter and allowing it to cool slightly. In a mixing bowl, combine 1 cup of granulated sugar with the zest of one lemon. Mix in melted butter, 3 tablespoons of oil, 1 large egg, 1 egg yolk, 2 tablespoons of fresh lemon juice, and 1 teaspoon of vanilla extract until lightened in color. Fold in 2½ cups of flour, 1 tablespoon of cornstarch, ½ teaspoon of baking powder, ¼ teaspoon of baking soda, and ¼ teaspoon of salt. Chill for at least 30 minutes.

Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper. Scoop about 3 tablespoons of the dough per cookie onto the sheets and flatten slightly. Bake for 12-13 minutes until the edges are golden and centers aren't wet. Cool on a wire rack.

While the cookies cool, grind 1½ cups of freeze-dried strawberries into a fine powder. Cream 1 cup of butter for 2 minutes, then gradually mix in the ground strawberries, 2½ cups of powdered sugar, ¼ cup of heavy cream, 1 tablespoon of fresh lemon juice, 1 teaspoon of vanilla extract, and a pinch of salt until silky.

Frost cooled cookies with the strawberry buttercream using a spatula and sprinkle with crushed freeze-dried strawberries and lemon zest if desired.