Start by cooking down the strawberries for the frosting so it has plenty of time to cool. In a small saucepan over medium heat, cook the strawberries stirring often until you have a thick strawberry sauce. This will take about 10 minutes and you should have about ¼ cup of the sauce when it's done. Set aside to cool completely.
To make the sugar cookie dough, beat the butter, cream cheese, and sugar on medium speed until light and fluffy (about 2 minutes). Add the egg and vanilla and beat until well incorporated. In a medium bowl, stir together the flour, cornstarch, baking powder and salt.
Gradually mix the flour mixture into the butter mixture on medium-low speed until well combined. Transfer the dough to the fridge to chill for 30 minutes. Meanwhile, preheat oven to 350℉ and line a baking sheet with parchment paper.
Use an ice cream scoop to scoop out the chilled dough (about 3 tablespoons). Roll into a smooth ball and place onto the prepared baking sheet. Use your palm to slightly flatten the ball. Leave 2 inches of space between the cookies. Bake for about 12 minutes or until the tops of the cookies look matte. Do not overbake!
Let the cookies set for about 5 minutes before transferring to a cooling rack to cool completely.
To make the frosting, beat the butter in a medium size mixing bowl until lighter in color and fluffy (about 5 minutes). Add the cooled strawberry sauce and vanilla and beat until combined. Beat in the powdered sugar until smooth. Use a spoon or offset spatula to add a generous amount of frosting to each cookie.
Store in an airtight container in the refrigerator for up to 4 days. Let the cookies come to room temperature before enjoying.