Preheat oven to 375 F. Grease and line an 8x8 pan with parchment paper, then set aside.

In a large bowl, mix the blackberries, strawberries, 2 tablespoons of sugar, and cornstarch until combined, then set aside.

In another large bowl, combine the flour, oats, baking powder, and salt. Add the cold butter and, using your fingers, press the butter into the flour mixture so that it becomes like small crumbs/pebbles of butter within the flour mixture (much like you would do with pie dough). Add the egg and vanilla extract and, using a wooden spoon, mix until it forms together into a dough like wet sand.

Spread 2/3 of the dough mixture into the pan, then add the berries. Top with dots of the dough mixture (it won't be able to cover all the berries, and this is ok). Sprinkle on top more old fashioned oats and turbinado sugar, then place in the oven and bake for 40-45 minutes, until the top is golden brown and the juices from the berries starts to subside a little.

Remove from the oven and let cool before cutting.