To make the crust: In a large bowl, combine the flour, thyme, and salt. Grate the butter into the mixture. Using your hands or a pastry blender, work the butter into the flour mixture until the pieces are about the size of peas. Add the milk, ¼ cup at a time, and blend until a soft, moist dough is formed. Add a few more tablespoons of milk if the dough seems dry. It should be somewhat sticky.
Form the dough into a ball, then flatten into a round disk. Wrap in plastic wrap and refrigerate for at least 1 hour.
To make the topping: In a medium bowl, combine the flour, brown sugar, cheese, and pepper. Grate the butter into the mixture. Using your hands or a pastry blender, work the butter into the flour mixture until the pieces are about the size of peas. Refrigerate until ready to use.
Preheat the oven to 350°F. When the piecrust dough is chilled, place it on a well-floured surface and roll to about a 1/8-inch thickness, turning and flouring the dough often to keep it from sticking. Fold the dough in half and place in a 9- or 10-inch pie plate. Unfold the dough, turn under the edges, and crimp them.
To make the filling: In a large bowl, combine the apples, brown sugar, cornstarch, and nutmeg. Transfer to the prepared crust.
Crumble the crumb topping over the apples. Bake for 1 hour, or until the crust is browned and the apples are soft. Place on a rack to cool for at least 30 minutes before slicing.