1. Heat the oven to 350 degrees. Use nonstick cooking oil spray to grease the wells of one 12-cup or two 6-cup muffin pans.
2. Whisk together the flour, sugar, baking powder, baking soda, salt, cinnamon and allspice in a mixing bowl.
3. Whisk together the oil, eggs, squash and shredded apple in a separate mixing bowl. Gently fold the dry ingredients into the wet mixture, taking care not to overmix. Divide the batter evenly among the muffin wells. Bake for 25 minutes or until a cake tester inserted into the center of the one of the muffins comes out clean.
4. Transfer the muffin pan(s) to a wire rack to cool for 10 minutes before dislodging the muffins. Serve warm or at room temperature, with butter or apple butter.
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