Preheat oven to 325 degrees. Butter and flour a 5-by-9-inch loaf pan.
In a medium bowl, whisk together all-purpose flour, whole-wheat flour, baking powder, kosher salt, cinnamon, nutmeg, cloves, and allspice.
Use an electric mixer to beat together butter and light brown sugar until fluffy, about 4 minutes. Add molasses; beat until combined. Add eggs, 1 at a time, plus vanilla, and beat until incorporated.
With the mixer on low, add half the flour mixture and beat until just combined. Beat in applesauce; add remaining flour mixture until just incorporated.
Peel and core apples, reserving the peel and the core. Cut apple into 1/4-inch cubes, and fold into the batter. Transfer to pan, and bake until a toothpick inserted into the center comes out clean, about 1 hour, 10 minutes. Transfer to a wire rack; let cool completely.
In a small saucepan, combine granulated sugar, 1/2 cup water, and apple core and skin. Bring to a simmer and cook over medium-high heat until syrupy, about 10 minutes.
Remove cake from pan, and brush glaze over top and sides of cooled cake.