1.To Prepare: Using a blender, food processor, or immersion blender, purée blackberries until totally smooth. If seeds bother you, the purée can be strained using a fine-mesh sieve; if you do plan to strain, add a couple extra blackberries to make up for the loss. Measure out exactly 6 ounces fresh blackberry purée (about 3/4 cup; 170g) and bring to about 70°F (21°C) before using.
2. Adjust oven rack to lower-middle position and preheat to 350°F (180°C). Lightly grease an 8- by 2-inch or 8- by 3-inch anodized aluminum cake pan and line with parchment (explanation and tutorial here).
3. To Make the Cake: In the bowl of a stand mixer fitted with paddle attachment (see note), combine butter, sugar, baking powder, baking soda, salt, and cinnamon. Mix on low speed to roughly incorporate, then increase to medium and beat until fluffy and light, about 5 minutes. About halfway through, pause to scrape bowl and beater with a flexible spatula.
4.With mixer still running, add egg whites a little at a time, letting each addition fully incorporate before proceeding to the next. Reduce speed to low and sprinkle in about one-third of cake flour, followed by one-third of prepared blackberry purée. Repeat with remaining flour and fruit, working in thirds as before.
5.Scrape bowl and beater with flexible spatula and resume mixing on medium speed for about 3 seconds to ensure everything is well combined. The batter should look bright purple and register between 65 and 68°F (18 and 20°C) on a digital thermometer. (Significant deviation indicates ingredients were too warm or too cold, which can lead to textural problems with the cake.) If the batter looks streaky or tie-dyed, gently fold with a flexible spatula until streaks disappear.
6.Scrape into prepared pan, spread into an even layer, and bake until puffed and firm, about 30 minutes. Cool directly in pan for 1 hour, then run a butter knife around the edges to loosen. Invert onto a wire rack, peel off parchment, and place on a serving plate until fully cool, about 1 hour. Top with a batch of Cream Cheese Frosting and a handful of fresh blackberries and cut with a chef's knife to serve. Wrapped in plastic, this cake will keep for up to 3 days at cool room temperature.