For the Ice Cream Filling:
Soften ice cream in a large mixing bowl at room temperature for 15 to 20 minutes, stirring occasionally. Mash strawberries with a fork; add sugar and stir until well blended. Mix ice cream and strawberries until well blended. Line two 9 x 9-inch square pans with plastic wrap and spread an equal amount of ice cream mixture in each. Mixture will be about 3/4-inch thick. Press another layer of plastic wrap on top of ice cream and freeze for at least 2 hours.
For the Cookies:
Combine flour, cocoa, baking powder and salt in a bowl until well blended; set aside.
Beat sugar, oil, eggs and vanilla in large bowl with electric mixer for 1 minute or until well mixed. Add flour mixture at medium speed for 1 to 2 minutes until well blended.
Divide dough in half. Form each half of dough into a 9 x 9-inch square on parchment or wax paper. Cover squares with another layer of paper and freeze for at least 2 hours or up to 1 month. (Dough should fit easily on top of prepared ice cream).
To bake cookies, preheat oven to 350°F. Remove dough from freezer; thaw briefly for 5 minutes, then roll to smooth and flatten if needed. Cut dough into 3 x 3-inch squares and place 1 inch apart on parchment paper lined baking sheets. Bake 10 to 12 minutes until cookies are set and bottoms are lightly browned. Cool cookies for 1 to 2 minutes on baking sheet; transfer to cooling rack to cool completely.
To assemble ice cream sandwiches:
Cut ice cream into 3 x 3-inch squares. Place ice cream on bottom of one cookie and top with a second cookie. Wrap in foil and keep frozen until ready to serve.