Preheat the oven to 350 degrees F, with the rack set in the lower third of the oven. Line 2 baking sheets with parchment paper.
In a medium bowl, whisk together the all-purpose flour, whole wheat flour, baking powder, baking soda, salt, cinnamon and nutmeg. Set aside.
In a large bowl with a hand-held mixer (or use a stand mixer), beat the butter and sugar until the mixture is light and fluffy, 4 to 5 minutes. Add the applesauce and canola oil, and mix again.
Add the egg and vanilla, and beat until combined.
With the mixer set on low, gradually add the flour mixture to the butter mixture, and beat until just combined. Do not over beat.
Stir in the oats, chocolate chips and dried cherries.
Using 1½ tablespoon for each cookie, scoop out dough, then roll into a ball between your hands. Place the cookies on the prepared baking sheets, 2 inches apart, and flatten slightly.
Bake until the edges are set and just starting to brown, and the centers are still soft, 10 to 12 minutes. Let the cookies to cool on the baking sheet for 10 minutes, then transfer to a wire rack to cool.