Preheat oven to 350 degrees F. Line a 9 by 13-inch pan with non-stick foil.
Cream butter and cream cheese together (it will look a bit curdled). Add egg and mix until smooth. Combine creamed mixture with dry brownie mix, folding and pressing batter by hand until mixed. Mixture will be thick. Spread evenly in foil-lined pan and bake for 15 minutes. Cool to room temperature.
Beat cream cheese, 3/4 cup sugar, and vanilla with electric mixer until well blended. Add eggs, 1 at a time, mixing on low speed after each addition just until blended. Stir strawberry puree into batter.
Spoon strawberry batter over pre-baked, cooled brownie bottom.
Bake for 40 to 45 minutes or until center is set and edges are lightly browned. Cool to room temperature, then refrigerate while preparing topping.
Combine boiling water and both boxes of gelatin. Stir until no more sugar or gelatin granules are visible and liquid is clear. Cool to lukewarm temperature. Add strawberry chunks to the gelatin and mix well.
Pour cooled strawberry gelatin mixture over top of cooled cheesecake and spread evenly. Refrigerate for 2 hours or until topping is firm. Cut into small bars before serving. Refrigerate leftovers.