1. Make the tart shell: in a medium bowl, combine the flour with the salt. Using a pastry blender or 2 knives, cut in the butter until the mixture resembles coarse meal.
2. Add the egg and water and mix with a fork until a dough forms.
3. Turn the dough out onto a lightly floured work surface and pat it into a disk. Wrap the dough in plastic and refrigerate until firm, at least 1 hour or overnight.
4. On a lightly floured surface, roll out the dough to a 13-inch round. Fold the dough in half and carefully transfer it to an 11-inch tart pan with a removable bottom. Unfold the dough and gently press it into the pan without stretching. Trim the overhang. Wrap the tart shell in plastic and refrigerate it until firm, about 1 hour, or overnight.
5. Preheat the oven to 350°F.
6. Line the tart shell with foil and fill with pie weights, dried beans or rice. Bake the shell for about 1 hour, or until lightly browned around the edge. Remove the pie weights and foil and bake the shell for 10 minutes, or until evenly browned and cooked through. Transfer to a rack to cool. Leave the oven on.
7. Make the filling: in a medium saucepan of boiling salted water, cook the asparagus until tender, about 4 minutes. Drain and rinse under cold running water. Drain again and pat dry with paper towels.
8. Arrange the asparagus in the tart shell with all the spears facing the same direction; trim them to fit if necessary.
9. In a medium bowl, whisk the eggs to mix. Beat in the milk, melted butter and salt. Pour the custard mixture over the asparagus and sprinkle the Parmigiano-Reggiano over the top.
10. Set the tart on a baking sheet and bake for about 45 minutes, or until the custard is just set. Transfer the tart to a rack to cool for about 10 minutes, then cut into 8 wedges and serve warm or at room temperature.

Serves 8.

Make ahead: the tart shell can be baked, cooled, and kept well wrapped in its pan at room temperature for up to 1 day.