Heat the oven to 350°F. Line a 9x13-inch pan with parchment, leaving overhang on two sides.
In a large bowl, whisk together the flour, oats, brown sugar, baking powder, salt, and cinnamon. Pour in the melted butter and vanilla and stir until crumbly and evenly moistened.
Press about two-thirds of the mixture firmly and evenly into the bottom of the pan. Bake for 15 minutes, until just set at the edges.
Meanwhile, toss the sliced peaches with the granulated sugar, cornstarch, and lemon juice until every slice is coated.
Spread the peaches evenly over the hot par-baked crust. Squeeze the remaining dough into clumps and scatter them over the fruit, leaving some gaps so the filling can bubble through.
Bake 40 to 45 minutes more, until the topping is deep golden and the filling bubbles at the edges. Cool completely in the pan, at least 2 hours, before lifting out and slicing into 16 bars.