Preheat the oven to 350ºF. Spray a 8x4 inch loaf pan with nonstick spray then line with parchment paper, allowing some parchment overhang. Set aside.
In a large bowl, place the granulated sugar and with your fingers, rub in 1 teaspoon of lemon zest to release the oils. Set aside.
1 cup granulated sugar, 1 tablespoon lemon zest
In the bowl of a mixer, add the butter and lemon sugar, beating on medium speed until light and fluffy, scraping down the sides as needed.
¾ cup unsalted butter
Add eggs one at a time, allowing each to incorporate fully before adding the next. Once all are incorporated, beat on medium speed for 1 minute until light in color.
3 large eggs
Mix in vanilla and lemon juice. Mixture may look curdled.
1 tablespoon freshly squeezed lemon juice , 1 teaspoon vanilla extract
With the mixer off, add in all-purpose flour, cake flour, baking powder, baking soda, and salt. Mix on low speed just until incorporated. Do not overmix.
1 cup all-purpose flour, ½ cup cake flour, 1 ½ teaspoons baking powder, ½ teaspoon baking soda, ½ teaspoon kosher salt
With the mixer off, add the sour cream (or Greek yogurt). Turn mixer to low speed and mix just until incorporated. Scrape the bottom and sides of the mixing bowl as needed to ensure the ingredients are evenly incorporated.
1 cup full-fat sour cream
Add in the remaining 2 teaspoons lemon zest and stir on low just until incorporated. Do not overmix.
1 tablespoon lemon zest
Spread about half the batter (2 cups) into prepared pan. Top with half of the jam, scooping large dollops on top of the batter. Use a butter knife to gently swirl together the jam and batter. Don’t go overboard with the swirling. Repeat this process - top with remaining batter then the remaining jam. Swirl again. ½ cup seedless blackberry jam
Bake in the preheated oven for 30 minutes then cover loosely with foil to prevent the top from over browning and bake for another 30 minutes or until a toothpick inserted into the center comes out clean.
Allow to cool in the pan for 10 minutes, then gently lift out of the pan using the parchment overhang and allow to cool on a wire rack set over a baking sheet until completely cool. The baking sheet will catch any excess glaze when poured on top of the cake.
Make the lemon glaze by whisking together the powdered sugar and lemon juice a little at a time, until desired thickness is achieved. You may not use all the lemon juice.
1 ¼ cups powdered sugar, 2 - 4 tablespoons freshly squeezed lemon juice
Just before serving, pour the glaze over the cooled cake, slice, and serve.