Combine the sugar and lemon zest with your fingers, rubbing to release the oils in the zest.

In a large bowl, whisk together the melted butter, lemon sugar, oil, eggs, lemon juice, and vanilla. Continue whisking until the ingredients are incorporated and the mixture lightens in color, about 1 minute.

Add flour, cornstarch, baking soda, baking powder, and salt to the bowl. Switch to a rubber spatula and mix in the dry ingredients until everything is fully incorporated.

Cover the dough with plastic wrap and chill in the fridge for 30 minutes.

Meanwhile, line 2 baking sheets with parchment paper or a silicone mat and preheat the oven to 350°F.

Once the dough has chilled, use a #24 cookie scoop (about 3 Tablespoons) to portion the dough. Roll each scoop into a ball and place them onto the prepared baking sheets, spacing them at least 2” apart. Press each dough ball down with your palm to flatten it slightly. Keep the remaining dough that isn’t baking in the fridge.

Bake at 350°F for 12-13 minutes or until the centers of the cookies no longer look wet and the edges start to turn golden brown. Allow to cool completely before frosting.

In the bowl of a stand mixer fitted with a paddle attachment, cream the butter on medium speed until smooth, about 2 minutes. Scrape down the bowl with a rubber spatula.

Add the ground freeze-dried strawberries, powdered sugar, heavy cream, lemon juice, vanilla, and salt, to the bowl. Start mixing on low speed until the ingredients are combined, so the powdered sugar doesn’t spill out. Scrape the bowl well with a rubber spatula.

Increase to medium speed and mix for 2 minutes. The frosting should be smooth and creamy.