Preheat oven to 425 degrees F (220 degrees C). Line a baking sheet with aluminum foil and grease lightly with cooking spray.
Place kabocha squash skin-side down on the baking sheet.
Roast in the preheated oven until fork-tender, 30 to 40 minutes. Cool until easily handled, about 10 minutes. Scoop squash flesh into a food processor; puree until smooth. Discard skin.
Reduce oven temperature to 350 degrees F (175 degrees C). Grease 2 mini muffin tins with cooking spray.
Stir white sugar, brown sugar, canola oil, and olive oil in a large bowl until smooth. Beat in eggs, 1 at a time. Stir in 3/4 cup pureed squash and vanilla extract.
Mix whole wheat flour, all-purpose flour, cinnamon, baking powder, baking soda, and salt together in a bowl. Stir gradually into the sugar mixture, alternating with apple cider, until smooth. Pour into muffin tins, filling each cup 3/4 full.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 10 minutes.