Preheat oven to 350°F (177°C). Generously grease 2 mini bundt cake pans.
Whisk together flour, baking powder, baking soda, and salt in a medium bowl. Set aside.
In the bowl of a stand mixer, beat butter and 1 ¼ cups granulated sugar on high until light and fluffy.
Reduce speed to low and mix in yogurt, lemon juice, and lemon zest. (The batter will look loose and have a lot of liquid, so it’s important to mix on low).
Add the eggs and continue mixing. The batter may appear curdled, but will bake into soft, moist cakes.
Add the flour mixture and mix at low speed until just combined. Let the batter rest for 5 minutes.
Spoon batter into the prepared pans with a cookie scoop, filling each cavity three-quarters full.
Bake for 18-22 minutes until the mini bundt cakes are puffed up and golden around the edges.
Cool bundt cakes in the pan for 5 minutes, then carefully invert them onto a wire rack to cool completely.
For the glaze: combine blackberries, ¼ cup granulated sugar, and water in a small saucepan over medium-high heat.
Bring to a boil, then reduce the heat to a simmer for 4 or 5 minutes, or until slightly thickened.
Strain the mixture through a fine-mesh sieve, discarding the pulp.
In a small bowl, whisk together 2 ½ cups powdered sugar, 1 tablespoon heavy cream, and 2 tablespoons of the blackberry juice until smooth and glossy. Adjust with more powdered sugar (for thicker glaze) or blackberry juice (for thinner).