Position a rack in the center of the oven and heat it to 375°F (190°C). Lightly butter your baking dish and set it on a baking sheet.
Peel, core, and slice the apples into 1/4 inch (6 mm) pieces, keeping them roughly the same thickness so they cook through evenly.
Toss the apple slices with the cornstarch, cinnamon, and nutmeg, then add the brown sugar, lemon juice, vanilla, and salt and stir until glossy. Transfer to the prepared dish and spread evenly.
Whisk the flour, brown sugar, granulated sugar, cinnamon, and salt in a second bowl. Cut in the cold cubed butter until the mixture forms clumps the size of small peas, stopping while some pieces are still chunky.
Spoon the topping evenly over the apples, covering edge to edge, and leave the surface craggy rather than pressing it flat. Sprinkle the turbinado sugar over the top.
Bake for 40 to 45 minutes, until the topping is deep golden and the fruit juices bubble up around the edges. Tent loosely with foil for the last 10 minutes if the top browns too fast.
Let the crumble rest for 15 to 20 minutes so the cornstarch sets the juices, then serve warm on its own or with a scoop of ice cream.