Preheat your oven to 350°F and grease or line a standard loaf pan with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt.
In a large bowl, whisk the sugar and eggs until smooth, then mix in the yogurt, oil, lemon zest, lemon juice, and vanilla extract until fully combined.
Add the dry ingredients to the wet ingredients and stir just until no dry streaks remain.
Toss the blackberries with 1 tablespoon of flour, then gently fold them into the batter.
Pour the batter into the prepared loaf pan and smooth the top.
Bake for 50–60 minutes, or until a toothpick inserted in the center comes out clean. If the top browns too quickly, loosely cover with foil during the last 10–15 minutes.
Let the loaf cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.
If using the glaze, whisk everything together starting with 1 tablespoon lemon juice.
Add more juice a few drops at a time until it becomes thick but spreadable—it should slowly mound rather than run.
Spread over the completely cooled loaf with a spoon or spatula.
Let sit 15–20 minutes to set before slicing.