Arrange a rack in the middle of the oven and heat to 350°F. Coat a 9x9-inch or 11x7-inch glass baking dish with butter; set aside.
Whisk together the sugar and cornstarch in a medium mixing bowl until lump-free. Add the fruit and lemon juice and toss gently to coat. Transfer to the baking dish.
Topping: Combine the oats, flour, sugar, and salt in a large bowl and stir until combined. Drizzle the butter and vanilla over the oat mixture and stir to combine. Set aside or store in the freezer in a zip-top freezer bag for later use.
Scatter the crisp topping evenly over the fruit mixture, leaving large clumps intact. Bake until the fruit juices are bubbling around the edges of the baking dish and the topping is golden and firm to the touch, 30 to 35 minutes.
Let the crisp cool on a wire rack for at least 15 minutes before serving. If transporting to a picnic or party, let the crisp cool completely to give the fruit filling time to set. Crisps will keep, covered and refrigerated, for up to 1 week. Serve cold, room temperature, or re-warmed in a low oven for 20 minutes.
Make ahead: Crisp topping can be made and frozen up to 3 months in advance.
Storage: Cover and refrigerate leftover crisp for up to 3 days. Enjoy cold for breakfast or gently rewarm in a low oven for 20 minutes before serving.