Preheat the oven to 350°F. Butter a 9×9-inch baking dish and line it with parchment paper.
In a large bowl, whisk together the flour, baking soda, and salt. Add the oats, brown sugar, and granulated sugar, whisking and breaking up any brown sugar clumps.
Stir the vanilla into the melted butter, then pour it over the oat mixture and stir until evenly moistened.
Press two-thirds of the crumble mixture into the prepared dish and set the remaining third aside.
In a medium bowl, toss the diced strawberries and rhubarb with the lemon juice.
In a small bowl, whisk together the 1/3 cup sugar and cornstarch, then pour over the fruit and toss to coat.
Pour the fruit mixture evenly over the bottom crumb layer, then sprinkle the reserved crumble over the top.
Bake for 45 to 50 minutes, until the top is golden brown and crisp and the filling is bubbling.
Cool on a wire rack until warm, then cut into squares. Serve warm with vanilla ice cream if desired.