Mix the flour, butter, sugar, and salt quickly with your fingertips until the butter is in pieces about the size of dried beans. Quickly stir the water into the mixture and then gather up the dough onto a sheet of plastic wrap and press it into a flat dish shape. Refrigerate dough for 1 hour.
Line a cookie sheet or jelly roll pan with parchment or a silicon pad. Roll the dough out onto a lightly floured surface into a rectangle about 15-inches long by 10-inches wide, with rough edges. Roll the dough up onto your rolling pin and then unroll it directly onto the lined pan.
Roast the hazelnuts on a cookie sheet for 7 or 8 minutes. Transfer the nuts to a food processor, add the 1/4 cup sugar, egg, 1 tablespoon butter, and vanilla, and process until a creamy paste forms. Spread the frangipane on the dough, stopping about 1 1/2-inches from the outer edge on all sides.
Peel the apples, remove the cores, and cut them into 1/2-inch thick slices. Arrange them attractively in a slightly overlapping pattern, like the tiles of a roof, on the pastry dough so that the apples cover the frangipane. Fold the edge of the dough back over the apples to make a 1 to 1 1/2-inch border of dough. Increase the oven temperature to 400 degrees F. Sprinkle 2 tablespoons sugar over the apples, letting a little of the sugar fall onto the border to help crystallize the dough. Break the remaining 2 tablespoons butter into pieces and scatter them over the tart. Bake for 1 hour. The dough should be nicely crystallized all over.
Remove the finished tart from the oven and cool on a rack until lukewarm. Dilute the apricot preserves with the calvados or water and glaze the surface of the tart, spreading it carefully on the apples with the underside of a spoon. Serve at room temperature cut into little wedges, with plain cream, sour cream, creme fraiche, or ice cream, if desired.