Preheat the oven to 350°F (175°C). Grease a 9x9-inch baking pan or line it with parchment paper for easy removal.
In a medium bowl, combine the sliced strawberries, chopped rhubarb, 1/4 cup of granulated sugar, cornstarch, 1 teaspoon of vanilla extract, and lemon juice. Stir well and set aside to macerate for about 15 minutes.
In another bowl, mix together the flour, rolled oats, brown sugar, cinnamon, and salt. Pour in the melted butter and stir until the mixture is crumbly.
Press half of the crumb mixture into the bottom of the prepared baking pan to form the crust. Bake for 10 minutes, then remove from the oven.
While the crust is baking, prepare the cheesecake layer. In a mixing bowl, beat the softened cream cheese with the remaining 3/4 cup of granulated sugar until smooth. Add the powdered sugar, egg, and 1 teaspoon of vanilla extract, mixing until fully combined.
Spread the cheesecake mixture evenly over the baked crust. Spoon the strawberry-rhubarb mixture on top of the cheesecake layer, spreading it out gently.
Sprinkle the remaining crumb mixture over the top of the fruit layer.
Bake for an additional 30-35 minutes, or until the topping is golden brown and the cheesecake is set. Allow to cool at room temperature for at least 1 hour, then refrigerate for at least 2 hours before cutting into bars.
Serve chilled or at room temperature.