How to Make It
In a bowl, stir the 1 1/3 cups flour with the sugar and salt. Using your fingers, rub the butter and shortening into the flour until the mixture resembles coarse crumbs. Sprinkle the ice water on top and stir just until combined. If the dough is sticky, work in 1 more tablespoon of flour. Shape the dough into a ball and pat it out to a 6-inch disk. Wrap in wax paper and refrigerate for at least 1 hour or overnight.
Preheat the oven to 400°. On a lightly floured surface, roll out the dough to a 13-inch circle. Transfer the dough to an 11-by-1-inch tart pan with a removable bottom. Trim the overhang flush with the rim and prick the bottom all over with a fork. Butter a large piece of foil, set it buttered side down on the dough and fill with rice or pie weights. Bake for about 20 minutes, or until the edges are golden. Remove the foil and weights and bake for about 5 minutes longer until the bottom is just cooked through. Let cool slightly. Lower the oven temperature to 375°.
In a food processor, pulse the walnuts until finely ground but not pasty. In a medium bowl, beat the butter with 1/3 cup of the sugar. Beat in the egg, rum and flour until blended. Fold in the walnuts; the mixture may look curdled, but it won't affect the tart.
Peel, halve and core the apples and cut them lengthwise into 1/8-inch slices. Transfer to a medium bowl and toss with the lemon juice, cinnamon and nutmeg.
Spread the walnut mixture evenly in the tart shell. Arrange the apple slices on top in concentric circles and sprinkle with the remaining 2 tablespoons sugar. Bake the tart for about 35 minutes, or until the apples are browned at the edges and tender when pierced with a knife. Let cool slightly on a rack.
In a small saucepan, melt the apple jelly over low heat, stirring, until smooth. (If using apricot jam, strain it after melting.) Brush the apples with the jelly. (The tart can be kept at room temperature for up to 4 hours. It can be rewarmed gently in a 325° oven.) Remove the sides of the pan and serve the tart warm.