Preheat oven to 350°F with rack in middle. Spray the bottom and sides of an 8* or 9-inch round cake pan with cooking spray - one with flour, if possible. Set aside.
Sift flour, baking powder, baking soda, together into a medium bowl.
In a large bowl or the bowl of a stand mixer, beat the butter, sugar, and 2 eggs until very light in color and fluffy, about 2-3 minutes on high speed. Add the lemon extract and Greek yogurt and continue beating until very creamy and light in color, about 1 more minute.
With the mixer on low speed, slowly add in the flour mixture, until combined. Do not overmix.
Transfer the cake batter to the prepared cake pan. Top with blackberries and lemon zest, then sprinkly with sugar.
Bake until cake turns golden, and a toothpick inserted in the middle comes out clean, 40 minutues to 1 hour, depending on your oven.
Remove from oven and allow to cool completely in pan, on a baking rack. Once cooled**, you can invert the cake onto a plate. You can then gently slide your hands underneath and turn it right-side up onto the plate. Or - place another plate on top (or bottom, rather) and flip it over using both plates, if you're worried it may break if flipping with just your hands.