Combine all dry ingredients in a medium bowl. In a large bowl, thoroughly whisk together eggs, applesauce, molasses, and oil. Fold dry ingredients into wet until no dry flour is visible.
Pour batter into the prepared loaf pan. Bake until a toothpick inserted into the center comes out clean, about 1 hour. Let cool for 30 minutes before removing from the loaf pan.
To ice, whisk together powdered sugar and enough water to make a thick yet pourable mixture. Drizzle over the top of the cake and let sit for a few minutes to harden.