Preheat oven to 425 degrees F (220 degrees C).

Mix 2 1/2 cups of flour and the salt in a bowl; cut the shortening into the flour mixture with a pastry cutter until the mixture is crumbly. Using a fork, mix in water, 1 tablespoon at a time, until the dough just holds together. Divide the dough in half and form each half into a ball. Working on a floured work surface, roll each ball out into a crust big enough for a 9-inch pie dish with an inch or so to spare. Carefully lift a crust, fold into quarters, position into pie dish, and unfold the crust. Set the other crust aside.

Mix the sugar, 1/4 cup flour, and cinnamon together in a large bowl until thoroughly combined; stir in the lemon juice and apricots. Spoon the apricot filling over the pie crust in the dish, and top with the reserved crust. Crimp the edges of the crusts together with a fork, cut away the excess crust, and cut slits into the pie to allow steam to escape. Cover the edges of the pie crust with strips of aluminum foil to prevent burning.

Bake on center rack of the preheated oven until the pie is browned and the filling is bubbling and thickened, 35 to 45 minutes. Remove to a rack to cool and sprinkle top with 1 teaspoon of sugar.