Preheat the oven to 350 degrees F.
For the crust, put the flour and 1/4 teaspoon salt in the bowl of a food processor fitted with the steel blade. Cut 12 tablespoons (1 1/2 sticks) of the butter into large dice, add to the bowl, and pulse until the butter is the size of peas. With the machine running, add the ice water all at once and process until the dough becomes crumbly. Don't overprocess. Dump the dough out on a floured board, gather it loosely into a ball, cover with plastic wrap, and refrigerate for 30 minutes.
Roll the dough on a well-floured board and fit it into a 9-inch tart pan with a removable sides, rolling the pin over the top to cut off the excess dough. Butter 1 side of a square of aluminum foil and fit it, butter side down, into the tart pan. Fill the foil with rice or beans. Bake for 20 minutes. Remove the beans and foil from the tart shell, prick the bottom all over with a fork, and bake for another 10 minutes.
Meanwhile, heat the remaining tablespoon of butter in a small pan and saute the shallots over low heat for 5 minutes, or until tender. Place the goat cheese in the bowl of the food processor and process until crumbly. Add the cream, eggs, basil, 1/4 teaspoon salt, and the pepper and process until blended.
Scatter the cooked shallots over the bottom of the tart shell. Pour the goat cheese mixture over the shallots to fill the shell (if the shell has shrunk, there may be leftover filling). Bake for 30 to 40 minutes, until the tart is firm when shaken and the top is lightly browned. Allow to cool for 10 minutes and serve hot or at room temperature.
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