Preheat and Prepare: Preheat your oven to 375F (190C). Lightly grease an 8X8 inch baking dish.
Make the Peach Filling: In a large bowl, gently toss the sliced peaches with the sugar, lemon juice, cornstarch, and cinnamon. Mix until the peaches are evenly coated. Pour this mixture into your prepared baking dish and spread it out evenly.
Make the Topping: In a separate medium bowl, whisk together the flour, rolled oats, brown sugar, cane or granulated sugar, cinnamon, and salt.
Cut in the COLD Butter: Into the dry ingredients using a pastry cutter or your fingertips until the mixture resembles coarse crumbs with some pea-sized butter pieces remaining.
Assemble the Cobbler: Sprinkle the oatmeal topping evenly over the peach filling.
Bake to Perfection: Bake for 35-45 minutes, or until the fruit filling is bubbly and the oatmeal topping is a deep golden brown and crisp.
Rest and Serve: Let the cobbler cool for at least 15-20 minutes before serving. This allows the filling to set up properly. It's absolutely divine served warm with a scoop of vanilla ice cream or whipping cream.