Heat the oven to 375°F.
Make the topping:
Combine the flour, both sugars, salt, and cinnamon in a medium bowl. Rub in the butter with your fingertips until it's well blended and the mixture crumbles coarsely; it should hold together when you pinch it. Add the hazelnuts. Refrigerate the topping until you're ready to use it.
Make the crisp:
Put the apples and blackberries in a large bowl. Taste the fruit and sprinkle on the sugar as needed (use 2 Tbs. sugar for very ripe fruit, or up to 1/3 cup for tarter, less-ripe fruit).
In a small dish, dissolve the cornstarch in the lemon juice. Gently toss the mixture with the fruit.
Pour the fruit into an 8- or 9-inch square (or similar-capacity) glass or ceramic baking dish. Set the pan on a baking sheet to catch overflowing juices. Top the fruit with half of the topping (keep the other half refrigerated) and bake for 20 minutes.
Sprinkle the remaining topping over the crisp and continue baking until the fruit is tender when pierced with a knife, the topping is crisp, and the juices are bubbling, another 25 to 35 minutes. Let cool for 20 to 30 minutes. Serve warm.