Place top oven rack in the center position and pre-heat oven to 350°F. Line a standard 8½” loaf pan with parchment paper cut to size with the longer edges hanging over the sides. Set aside.
Cream the butter and one cup sugar together with a hand or stand mixer in a large mixing bowl.
Add the vanilla extract, applesauce, lemon juice, and lemon zest and mix until thoroughly combined.
Add the eggs, one at a time, and beat until incorporated into the mixture.
Combine the flour, baking soda, and salt in a large bowl and stir to combine. Gradually add the flour mixture to the wet ingredients a little at a time until incorporated into the batter. Use a plastic spatula to scrape the sides of the bowl while mixing.
Once the wet and dry ingredients are just blended, carefully fold the raspberries into the batter with a plastic spatula. Pour the batter into the prepared loaf pan and spread evenly. Sprinkle the remaining granulated sugar evenly across the top.
Place the loaf pan into the pre-heated oven and bake for 40-50 minutes, or until a toothpick inserted into the center of the loaf comes out clean.
Remove from oven and carefully lift the parchment paper to transfer the loaf to a wire rack to cool completely.
Prepare the lemon glaze by combining the powdered sugar, two tablespoons lemon juice, and the lemon zest together in a small bowl. Add additional lemon juice, if desired, to adjust the consistency.
Once cooled, drizzle the lemon glaze on top of the loaf cake right before serving. Enjoy!