Line a 9- by 5-inch loaf pan with waxed or parchment paper (or grease and flour it), then set it aside. Also grease a mini loaf pan. Heat the oven to 350°F.
Combine the applesauce, sugar, brown sugar, eggs, oil, yogurt, and vanilla extract in a large bowl. Whisk well to blend.
In another large bowl, combine the flour, baking powder, baking soda, salt, cinnamon, and nutmeg. Make a well in the dry ingredients with a wooden spoon and pour in the applesauce mixture. Stir just enough to combine; do not overmix. Fold in the apples and walnuts.
Scrape the batter into the prepared pans and smooth the top with a spoon. Evenly sprinkle one tablespoon of sugar over the top. Bake on the center oven rack about 30 to 35 minutes for the mini loaf and 50 to 55 minutes for the large loaf. (Loaf will register 200 degrees F on a cooking thermometer.)
Remove the pan from the oven and put it on a cooling rack for about 15 minutes, then remove the loaf from the pan and place it on the rack to finish cooling.
OPTIONAL GLAZE: Whisk 1/2 cup powdered sugar with 2-3 TBSP HOT water. Drizzle on cooled bread and let set.