Preheat oven to 400°F.
In a large mixing bowl combine the blackberries, blueberries, raspberries, sugar, flour, cornstarch, lemon juice, lemon zest, and vanilla. Toss to combine until evenly coated. Place in refrigerator until needed.
In a medium mixing bowl combine the oats, brown sugar, flour, cinnamon, and salt. Mix, then using a pastry blender, cut in the butter until pea-size clumps form. Place in refrigerator until needed.
Roll out or place a pre-made pie crust into a 9-inch pie dish. Press gently into the bottom of the pan, then flute the edge or crimp using a fork.
Add cold filling, then sprinkle on the topping. It will be a generous amount.
Place in the oven and bake for 30 minutes, then cover with foil and continue baking for 20-25 minutes. Tent the foil at the 45-minute mark to prevent the crust and topping from getting too brown.
Remove from the oven and allow the pie to cool at room temperature for 2½ hours before serving, or allow the pie to cool at room temperature for 1 hour, then 1 hour in the refrigerator before serving.