Preheat the oven to 350°F. Grease a 9x5-inch heavy-weight steel loaf pan or line it with parchment paper.
Add the sugar and lemon zest to a large bowl. Using clean hands, rub the lemon zest into the sugar until well incorporated. Add the lemon juice, oil, eggs, Greek yogurt, and vanilla extract. Whisk until smooth.
Add the 1 ¾ cups of the flour, baking powder, and salt, stirring gently until just combined, being careful not to overmix. It’s okay if there are a few lumps.
In a separate small bowl, toss the blackberries with the remaining 1 tablespoon of flour (to prevent sinking), leaving any excess flour in the small bowl.
Pour ⅓ of the batter into the prepared loaf pan, then gently layer in ½ of the blackberries, followed by ⅓ batter, then the remaining blackberries, then the remaining batter, smoothing the top.
Bake for 45-55 minutes, or until a toothpick inserted in the center comes out clean, being careful not to overbake.
Let cool in the pan for 10 minutes, then transfer to a wire rack to cool for an additional 10 minutes.
While the loaf cools, in a small bowl, make the lemon glaze by whisking together the powdered sugar and lemon juice until it’s completely smooth.
Drizzle the glaze over the top of the loaf, allowing some to drip down the sides. Allow to cool completely before slicing.