Preheat oven to 400 degrees Fahrenheit. Line bottom of a 9 inch non-stick pan with a sheet of parchment paper (cut to fit the bottom of the pan) then spray with cooking oil. Dust with flour, dumping out excess. Sprinkle 1 tbsp. sugar in pan and shake to distribute evenly (use 2 tbsp. of sugar if your berries are not sweet).
Arrange blackberries in a single layer in cake pan. Place the pan with blackberries in oven for 5 minutes to thoroughly warm the berries. Mix together flour, baking soda, baking powder, and salt in a bowl.
Beat together butter and remaining 1/2 cup sugar in a large bowl with an electric mixer at high speed until light and fluffy (about 2 minutes). Add egg, vanilla, and lemon peel, and mix until just incorporated.
Add buttermilk, mixing at a low speed until well incorporated, then add flour and mix at low speed until just combined.
Spoon batter evenly over berries and all the way to the edge of the pan with a rubber spatula, smoothing the surface, and bake in middle of oven until top is golden and a tester comes out clean (about 30 minutes).
Run a thin knife around the edge of the pan, then invert a large plate over pan and, using potholders to hold plate and pan together tightly, flip cake onto plate. Peel off parchment from cake if it sticks. Serve cake warm or cool.