Arrange your oven rack to the lower third of the oven. Preheat the oven to 350 degrees F. Spray two 5x8-inch loaf pans with nonstick spray.
PREPARE THE CAKE:
In a medium bowl, toss the pears with the white and brown sugars, cinnamon and nutmeg. Set aside. In a separate bowl, whisk together the flours, baking powder and salt.
In a large bowl, whisk together the sugar, eggs, milk, oil and applesauce. Whisk in the dry ingredients until they are just mixed in (don't over-mix). Stir in the pears and all of their sweet and spicy juices too. Divide the batter between the two pans. Bake 60 to 70 minutes, or until the loaves are baked through and test done (toothpick inserted in the centers comes out clean). Let the cakes cool for 15 minutes, then turn them out onto a rack to cool completely.
ADD THE ICING:
When cake is cool, prepare the icing. Melt the butter in a small saucepan over medium heat. Heat and stir for 5 to 7 minutes, or until the butter is lightly browned (you will see little flecks of brown throughout the melted butter when it is properly browned). Remove it from the stove and pour it into a bowl. Whisk in 1 cup of the powdered sugar. Add 3 teaspoons of the water and the vanilla. Whisk in more powdered sugar a little at a time until it thick but still pourable (add a teaspoon or two of additional water, if needed). Place paper towels under the rack and drizzle the icing onto the cakes. Serve immediately, or store in a covered container until ready to serve.