Preheat the oven to 400°F.
To make the filling, if using fresh peaches, bring a 4-quart pot of water to a boil and fill a large bowl with ice water. With a paring knife, slice through each peach skin from end to end, but leave the peach intact. Place the peaches in the boiling water for 30 seconds. Using a slotted spoon, transfer the peaches to the ice water for 30 seconds. Remove the skin from the peaches. Split the peaches in half, remove the pits, and slice each peach into 8 slices.
In a large bowl, toss together the sliced peaches, berries, lemon juice, cinnamon, and starch. Add the brown sugar, 1 teaspoon at a time, to taste depending on the sweetness of the fruit. Pour the peach mixture into an 8-inch square baking dish.
To make the topping, in a medium bowl, whisk together the flours, granulated sugar, baking powder, and salt. Add the milk, butter, and oil and stir to combine. Distribute spoonfuls of the batter over the peaches, leaving some patches uncovered. Sprinkle with the cinnamon. Bake until golden brown, 25-28 minutes.