Preheat oven and prep pan: Preheat oven to 350˚F. Brush an 8-by-8-inch cake pan with oil, line bottom and two sides with parchment, leaving a 2-inch overhang on sides, and brush parchment with oil.,

Grate squash and squeeze to remove moisture: Coarsely grate the squash on the large holes of a box grater (you should have 3 cups). Wrap squash in a clean kitchen towel and squeeze to remove as much moisture as possible.,

Combine dry ingredients; mix wet ingredients together: In a large bowl, whisk together flour, baking powder, five-spice, and salt. In another bowl, whisk together yogurt, eggs, butter, oil, granulated sugar, and vanilla.,

Add wet ingredients to dry; fold in squash and chocolate: Gently stir wet mixture into flour mixture just to combine but some bits of dry flour remain (do not overmix). Fold grated squash and chocolate into batter to evenly distribute.,

Transfer batter to pan and bake: Transfer to prepared pan and smooth top with an offset spatula. Bake until a toothpick inserted into center of cake comes out clean, 50 to 55 minutes.,

Cool: Let cool in pan on a wire rack 15 minutes. Remove cake from pan using parchment overhangs and let cool completely on rack.,

Make frosting and spread on cake: In a bowl, beat cream cheese and butter until smooth. Gradually add confectioners’ sugar, beating until light and fluffy. Spread evenly over top of cake and cut into squares to serve.