Preheat your oven to 350°F (175°C). Begin by preparing the pan: use 1 tablespoon of butter to thoroughly grease an 8-inch square pan, ensuring all edges and corners are covered.

Wash and dry the blackberries, then arrange them in a single layer at the bottom of the pan, covering the entire surface. Sprinkle 1 tablespoon of sugar evenly over the blackberries and set the pan aside.

In a large bowl, combine the sugar and lemon zest. Use your fingers to rub the lemon zest into the sugar to release the oils.

Add the softened butter to the bowl and mix with a stand mixer or hand mixer for 2-3 minutes until the mixture is fluffy and well combined.

Incorporate the eggs and vanilla extract, mixing until just combined, which should take about 1 minute.

In a separate bowl, whisk together the flour, baking powder, and salt.

Gradually add half of the dry ingredients to the butter mixture, mixing until just combined. Then, add half of the yogurt and mix again until just combined.

Repeat the process with the remaining dry ingredients and yogurt. The batter will be quite thick.

Transfer the batter into the prepared pan, spreading it evenly over the blackberries.

Bake in the preheated oven for 40-55 minutes, or until a toothpick inserted into the center comes out clean.

Once baked, allow the cake to cool for about 10 minutes before carefully flipping the pan onto a cooling rack. Gently lift the pan off to reveal the cake.

Let the cake cool for at least 20 minutes, or until it is completely cooled. Store any leftover cake in a sealable container.