Preheat oven to 325 degrees F.
Grease an 8 ½ x 4 ½-inch loaf pan with softened butter or nonstick spray.
In a medium bowl, whisk together flour, kosher salt, baking powder, and baking soda.
In a small bowl, whisk together lemon juice and greek yogurt.
In the bowl of your stand mixer fitted with the paddle attachment, beat butter, sugar, and lemon zest on medium speed until light and fluffy, about 3 to 5 minutes.
Beat in eggs, one at a time, until combined.
Add one third of the flour mixture, followed by half the yogurt mixture. Mix, then repeat again, ending with the remaining flour mixture. Mix again until just combined. Do not overmix.
Fold in blackberries with a rubber spatula until evenly distributed throughout the batter.
Transfer batter to prepared loaf pan. Smooth top with a rubber spatula or spoon. Top with extra blackberries, if desired.
Bake for about 65 to 75 minutes, or until the loaf is golden brown on top and a toothpick inserted into the center of the bread comes out with just a few crumbs. You might want to loosely tent foil over the top of the bread with about 20 minutes left to bake. This will prevent the sides from becoming too dark.
Let cool, uncovered, in pan for 10 minutes, then turn out onto rack to cool completely.
Make the glaze by combining confectioners sugar and lemon juice in a small bowl, mixing until thick, but easy to spread. Spread over top of the cooled bread.