Preheat your oven to 375°F. Line a jelly-roll pan with parchment paper, ensuring the paper extends slightly over the edges for easy removal.
In a medium bowl, sift together the all-purpose flour, spices, baking powder, baking soda, and salt. Set aside.
In a large bowl, use a handheld electric or stand mixer to beat the eggs with sugar until the mixture becomes thick and pale yellow in color. Then, fold in the kabocha squash purée into the egg and sugar mixture until well combined. Then, fold in the dry ingredients.
Pour the batter into the prepared jelly-roll pan and spread it evenly, smoothing the top with a spatula. Gently tap the baking pan on the counter to remove any bubbles.
Bake the sponge for 12 minutes, or until a toothpick inserted in the center comes out clean.
While the cake is baking, sift kinako onto a thin kitchen/tea towel. Ensure that the surface is just bigger than the cake and that the kinako is spread in a light, even layer. Next, sprinkle powdered sugar over the kinako. This combination will help prevent the cake from sticking when it is rolled.
While the cake is still hot, invert the cake onto the prepared towel and carefully peeling off the parchment paper. Starting from the short end, roll the cake up with the towel, creating a log shape. Place the rolled cake, seam side down, on a wire rack and let it cool completely.