Allowing the apples to render their juices with the sugar and eggs makes tender, flavorful muffins.
Position a rack in the center of the oven. Preheat the oven to 400°F. Grease a standard 12-muffin pan or line with paper cups.
Whisk together thoroughly:
      1 1/2 cups all-purpose flour
      2 teaspoons baking powder
      1 1/2 teaspoons ground cinnamon
      1 teaspoon baking soda
      Scant 1/2 teaspoon salt
Whisk together in a large bowl:
      2 large eggs
      3/4 cup sugar
Stir in and let stand for 10 minutes:
      1 1/2 cups packed coarsely grated or finely chopped peeled apples (1 medium apple yields about 3/4 cup), with juice
Stir in:
      5 tablespoons warm melted unsalted butter
      1/2 cup coarsely chopped walnuts or pecans
Add the flour mixture and fold just until the dry ingredients are moistened. Do not overmix; the batter should not be smooth. Divide the batter among the muffin cups. Bake until a toothpick inserted in 1 or 2 of the muffins comes out clean, 14 to 16 minutes. Let cool for 2 to 3 minutes before removing from the pan. If not serving hot, let cool on a rack. Serve as soon as possible, preferably the day they are baked.