Preheat your oven to 375°F. Spread oats and nuts on a baking sheet and toast them for 5–8 minutes until golden and fragrant. Set aside to cool.
Prepare two baking sheets by lightly coating them with cooking spray.
In a medium bowl, whisk together the flour, baking soda, baking powder, salt, 1/4 teaspoon cinnamon, and nutmeg.
In a large mixing bowl, combine the egg whites, shredded apple, brown sugar, apple butter, 3 tablespoons granulated sugar, oil, and vanilla extract. Stir until evenly blended.
Gradually add the dry ingredient mixture to the wet bowl, stirring until just combined. Fold in the toasted oats and nuts along with the diced dried apples.
Drop tablespoons of dough about 2 inches apart onto the prepared baking sheets.
Mix the remaining tablespoon of granulated sugar with the remaining 1/4 teaspoon cinnamon in a small bowl. Coat the bottom of a glass with cooking spray, dip it into the cinnamon sugar, and use it to flatten each cookie slightly. Re-dip into the cinnamon sugar as needed.
Bake each sheet for 10–12 minutes or until lightly browned around the edges. Allow to cool for 2 minutes on the pan, then transfer to wire racks to cool completely.