Preheat the oven to 350°F (175°C). Line a 9x13-inch baking pan with parchment paper, leaving overhang on the two long sides. Lightly grease any exposed corners.
In a large mixing bowl, whisk together the flour, rolled oats, brown sugar, granulated sugar, baking powder, salt, and cinnamon. Pour the melted butter over the dry ingredients and stir until the mixture looks like coarse, damp sand and clumps together when pressed.
Set aside about 1 ½ cups of the crumb mixture for the topping. Press the remaining crumb mixture firmly and evenly into the bottom of the prepared pan. Bake the crust alone for 12 minutes, until the edges look just barely set and lightly golden.
In a medium bowl, combine the sliced peaches, granulated sugar, cornstarch, lemon juice, cinnamon, vanilla extract, and a pinch of salt. Toss gently until every peach slice is coated and no dry cornstarch remains visible. Let the mixture sit for 5 minutes.
Spread the peach filling evenly over the par-baked crust, including any juice in the bowl. Sprinkle the reserved crumb topping evenly over the peaches, leaving some gaps. Do not press the topping down.
Bake at 350°F (175°C) for 30 to 35 minutes, until the topping is golden brown and the filling is bubbling at the edges. Tent with foil for the last 10 minutes if the topping browns too quickly.
Transfer the pan to a wire cooling rack and let the bars cool at room temperature for at least 1 hour. For the cleanest slices, refrigerate the cooled pan for 30 minutes to 1 hour before cutting.
Use the parchment overhang to lift the entire slab out of the pan and onto a cutting board. Slice into 16 even bars using a sharp knife, wiping the blade clean between cuts if the filling sticks.