Step 1Heat the oven to 350 degrees.
Step 2Peel the peaches: Bring a large pot of water to a boil. Cut a shallow X in the bottom of each peach, and blanch it quickly in the boiling water until you see the peel start to lift away from the peach. The blanching time required will vary depending on the ripeness of the peach. Transfer it to a bowl of ice water to stop the cooking, and peel away the skin with your fingers. If the peel proves difficult to remove, simply return the peach to the boiling water and try again. If after 5 minutes of blanching, the peaches are still not peeling easily, use a small paring knife to scrape away the skin.
Step 3Pit the peach by cutting it in half, following the cleft that runs down one side. Twist and the flesh will separate. Discard the pit and cut the peach into bite-size pieces. You should have about 7 to 8 cups of peeled, pitted, cut-up peaches.
Step 4Place the peaches in a bowl with the blackberries, cinnamon, brown sugar, 1 tablespoon flour and the lemon juice and stir gently to combine. Set aside while you make the topping.
Step 5In a food processor, briefly pulse together the sugar, salt and remaining 1 cup of flour just to combine. Distribute the chunks of butter over the top and pulse just until the mixture resembles wet, clumpy sand.
Step 6Dump the peach mixture into a 2-quart baking dish or gratin dish and give it a good shake to make sure it's evenly distributed. Spoon the crumbly mixture over the top, distributing it evenly. Bake until the top is crusty and brown and the peaches are soft, about 45 minutes.