Preheat the oven to 350°F (175°C) and line a 9x13 inch baking pan with parchment paper.
In a large mixing bowl, whisk together the flour, oats, brown sugar, baking powder, and salt.
Incorporate the chilled butter cubes into the dry mixture using a pastry blender or fork until the texture resembles coarse crumbs.
Press exactly half of the crumble mixture firmly into the bottom of the prepared baking pan to create an even base layer.
In a medium bowl, toss the rhubarb and strawberries with granulated sugar, cornstarch, lemon juice, and vanilla extract until the fruit is thoroughly coated.
Distribute the fruit filling evenly over the crust layer.
Sprinkle the remaining half of the crumble mixture over the top of the fruit, pressing lightly to adhere.
Bake for 45 to 50 minutes, or until the topping is golden brown and the fruit juices are bubbling at the edges.
Remove from the oven and allow the bars to cool completely in the pan for at least 2 hours to ensure the filling sets before slicing into squares.