Pre-heat oven to 350°F. Line an 8-½-inch loaf pan with parchment paper, letting the longer edges hang over the sides. Set aside.
Using a hand or stand mixer, beat together the butter and 1 cup of sugar in a large mixing bowl until creamy. Add the vanilla extract, applesauce, lemon zest, and lemon juice and blend until thoroughly combined.
½ cup unsalted butter, 1 cup granulated sugar, 2 teaspoons vanilla extract, ⅓ cup unsweetened applesauce, 1 Tablespoon lemon zest, 1 lemon, juiced
Incorporate the eggs one at a time, beating well after each addition until fully blended into the mixture.
3 large eggs
Stir together the flour, baking soda, and salt in a large bowl. Gradually add the flour mixture to the wet ingredients a little at a time until incorporated into the batter. Use a spatula to scrape down the sides of the bowl while mixing.
2 cups all-purpose flour, ½ teaspoon baking soda, ⅛ teaspoon salt
Fold the raspberries into the batter with a spatula. Pour the batter into the prepared loaf pan and spread evenly. Sprinkle the 2 tablespoon granulated sugar evenly across the top.
1 pint fresh raspberries, 2 Tablespoons granulated sugar
Place the loaf pan into the oven and bake for 40-50 minutes, or until a toothpick inserted into the center of the loaf comes out clean. (Check after 40 minutes and adjust final cook time accordingly).
Take the pan out of the oven and gently lift the parchment paper to move the loaf onto a wire rack to cool.
Make 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. Set aside.
¼ cup powdered sugar, ½ Tablespoon lemon zest, 1 lemon, juiced
Once the cake is completely cooled, drizzle the lemon glaze on top right before serving.