- Preheat oven to 350°F and line a 9 x 5-inch loaf pan with parchment paper, letting excess extend over sides of pan. Spray parchment paper lightly with nonstick spray.
- Whisk together flour, baking powder, baking soda, and salt in a medium bowl. Set aside 1 tablespoon of flour mixture to toss with blackberries.
- Place sugar and lemon zest in a food processor and pulse for 1 minute. Beat butter and lemon-sugar in a large mixer bowl at medium speed until light and fluffy; beat in egg and vanilla. Combine yogurt and buttermilk in a small bowl. Reduce speed to low and add half of flour mixture, beating just to combine. Add yogurt mixture, then finish with remaining flour mixture.
- Toss blackberries with reserved flour mixture and gently fold into batter. Pour batter into prepared pan and smooth top with an offset spatula.
- Bake for 55 to 60 minutes, until cake springs back to the touch and a toothpick inserted in the center comes out with a few crumbs attached. Cool in pan on a wire rack for 15 minutes; carefully lift cake from pan and cool completely.
- For the glaze, whisk together melted butter, salt, lemon zest, milk, and powdered sugar in a small bowl. Drizzle glaze over cake and and let set before serving.
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