Preheat oven to 425°F (218°C). Spray a 12-count muffin pan with nonstick spray or line with cupcake liners. Set aside.

In a large bowl, whisk the flour, baking powder, baking soda, cinnamon, allspice, and salt together until combined. Set aside. In a medium bowl, whisk the eggs, applesauce, maple syrup, oil, and vanilla together until combined. Pour the wet ingredients into the dry ingredients, stir a few times, and then add the shredded apple and pecans/walnuts. With a silicone spatula, fold everything together gently just until combined. The batter will be very thick.

Spoon the batter into liners, filling them all the way to the top. If using, sprinkle tops evenly with coarse sugar. Bake for 5 minutes at 425°F, then, keeping the muffins in the oven, reduce the oven temperature to 350°F (177°C). Bake for an additional 16–18 minutes or until a toothpick inserted in the center comes out clean. The total time these muffins take in the oven is about 21–23 minutes. (For mini muffins, bake 11–14 minutes at 350°F (177°C).)

Allow the muffins to cool for 5 minutes in the muffin pan, and then transfer to a wire rack to continue cooling, or enjoy warm.

Muffins stay fresh covered at room temperature for a few days, then transfer to the refrigerator for up to 1 week.