In a medium mixing bowl, use an electric mixer to beat butter until smooth. Add in egg and vanilla extract, mixing until incorporated. Add in strawberry cake mix and beat until well-combined. Chill in the refrigerator for at least one hour. Preheat the oven to 350F degrees. Line a baking sheet with parchment paper or a silicone mat. Using a cookie scoop or your hands, form chilled dough into balls about 2 inches wide. Place balls 2 inches apart on prepared baking sheet. Bake for 10 minutes, then remove from the oven. Let cool on the pan for a couple of minutes, then transfer to a wire rack to cool completely. Once cookie shells are cooled, prepare filling. In a medium mixing bowl, use an electric mixer to beat together pureed strawberries and cream cheese until smooth. Slowly mix in powdered sugar until desired consistency is achieved. Use a piping bag or knife to spread the frosting on the bottom of half of the cookies. Place another cookie on top of the frosting to make the sandwich.