Preheat your oven to 400°F (200°C). Line a muffin pan with paper liners.
In a small bowl, whisk together the flour, baking powder, baking soda, cinnamon, and salt.
In a large bowl, use an electric mixer to combine the eggs, sour cream, milk, granulated sugar, brown sugar, melted butter, and vanilla extract. The batter will be thick.
Gradually mix the dry ingredients into the wet ingredients using a silicone spoon or spatula. Be careful not to overmix—the batter should be lumpy. Lightly coat the blackberries in a small amount of flour, then gently fold them into the batter.
Divide the batter evenly among the muffin cups, filling each about ¾ full.
In a small bowl, mix the pecans, cinnamon, vanilla, brown sugar, flour, oatmeal, and melted butter until crumbly. Sprinkle the topping evenly over each muffin and press down gently.
Bake for 15-18 minutes, or until a toothpick inserted into the center of a muffin comes out clean. Baking time may vary slightly depending on your oven.
Let the muffins cool in the pan for 2-3 minutes, then transfer them to a wire rack. Serve warm or at room temperature.