How to Make It
In a food processor, pulse 2 1/2 cups of the flour and the salt. Add 2 sticks of the butter and pulse until it is the size of peas. Drizzle on the ice water and add the cheddar cheese and pulse until evenly moistened crumbs form; turn out onto a surface and form into a ball. Divide the dough in half. Flatten into disks, wrap in plastic and refrigerate until firm.
Preheat the oven to 375°. Set a baking sheet on the bottom rack. In a bowl, toss the apples, lemon juice, sugar, the remaining 1/4 cup of flour and the cinnamon.
On a floured surface, roll both disks of the dough to 13-inch rounds and transfer to parchment-lined baking sheets. Arrange the apples on top of each dough round, leaving a 2-inch border, and pour the liquid in the bowl over. Dot the apples with the butter and fold the edges of the rounds over the apples.
Bake the crostatas in the center of the oven for 1 hour and 10 minutes, until the crust is golden. Cover the edge of the crostatas if they begin to darken. Brush the apples with warm apricot preserves. Let the crostatas cool to room temperature, then serve.