*Used a combination of Granny Smiths (for their tart flavor and crisp texture) and Cortlands (for their sweetness and softer texture). If you use all Granny Smiths or all Cortlands (or other kinds of apples), adjust the amount of sugar to your taste.
In a medium-size mixing bowl, whisk together the flour and salt. Using an electric mixer, a pastry blender or fork, or your fingers, cut and mix the cheese and butter into the flour mixture until the fat and flour form a crumbly mixture. Add the lemon juice or vinegar, then sprinkle on just enough water so that you can gather the dough into a cohesive ball.
Divide the dough in half, and flatten each half into a 1-inch thick round. Wrap one of the rounds in plastic wrap or waxed paper, and refrigerate it while you work with the other. Transfer the remaining dough to a well-floured work surface. Roll it into a 12-inch circle, using as few strokes of the rolling pin as possible; the fewer times you touch the crust at this point, the more tender it'll be when it's baked. Transfer the circle of dough to a 9-inch pie plate (a giant spatula works great here), and gently fit it to the pan's contours. Again, if you push and stretch the dough too much during this stage, it'll shrink when you put it in the oven.
Peel, core and slice the apples. In a large, shallow frying pan, cook the apples with the salt, cinnamon, cider or juice, and sugar for 15 minutes, or until the apples are tender and the liquid is syrupy. Add the cooked sausage.
Spray the crust with some nonstick vegetable oil spray (this helps it stay crisp), then spoon the filling into the crust. Roll out the remaining crust, and center it over the filling. Press the edges of the top and bottom crust together, fold them under (onto the top of the pan's rim), and crimp.
The pie in a preheated 425°F oven for 15 minutes, then lower the heat to 375°F and bake for an additional 35 to 45 minutes, till the crust is very brown. Remove the pie from the oven, and serve it warm or at room temperature.