Preheat the oven to 375°F and line an 8-inch square baking pan with parchment paper, leaving a slight overhang for easy removal. Set aside.
Cream the butter and sugar in a large mixing bowl until light and fluffy, about 2–3 minutes. This step helps create a tender, bakery-style crumb.
Add the egg and vanilla extract, mixing until fully incorporated and smooth.
Whisk the dry ingredients together in a separate bowl, including the flour, baking powder, and salt.
Alternate adding the milk and dry ingredients to the butter mixture, starting and ending with the dry ingredients. Add half of the flour mixture and mix just until combined, then add half of the milk. Repeat with the remaining flour and milk, mixing gently to avoid overworking the batter.
Fold in the fresh blackberries carefully using a spatula, keeping them mostly whole to prevent excess juice from streaking the batter.
Spread the batter evenly into the prepared pan, smoothing the top.
Prepare the cinnamon sugar topping by mixing the granulated sugar, brown sugar, flour, cinnamon, and butter together until a crumbly streusel forms.
Sprinkle the streusel evenly over the top of the coffee cake batter.
Bake uncovered for 40–45 minutes, or until the top is golden and a toothpick inserted into the center comes out clean or with a few moist crumbs.
Cool completely in the pan before slicing and serving to allow the crumb to set.