Preheat the oven to 375°F. Lightly grease a 9×9-inch baking dish and set it aside.
Add the mixed berries, sugar, cornstarch, lemon juice, vanilla extract, and lemon zest to a large bowl. Gently toss until the berries are evenly coated.
Transfer the berry mixture to the prepared baking dish and spread it into an even layer.
In another bowl, combine the oats, flour, brown sugar, cinnamon, and salt. Add the softened butter and work it into the mixture with a fork or your fingertips until coarse crumbs form.
Sprinkle the crisp topping evenly over the berries, covering the surface without pressing it down.
Bake for about 40 minutes, or until the topping is golden brown and the berry filling is bubbling around the edges. Let it cool for several minutes before serving.