Start by peeling and slicing the peaches into even pieces. Place them in a large bowl and mix with lemon juice, brown sugar, cornstarch, vanilla, and cinnamon. Stir gently until the peaches are well coated and set aside.
In a separate bowl, combine flour, brown sugar, cinnamon, and salt. Add the cold cubed butter and cut it into the mixture using a pastry cutter or fork until it resembles coarse crumbs. Stir in the rolled oats until evenly mixed.
Spread the peach mixture evenly into the baking dish. Sprinkle the oat topping over the peaches in an even layer. Bake in a preheated oven until the topping is golden brown and the peach filling is bubbling around the edges. Let it cool slightly before serving to allow the filling to set.