In a heavy pot, bring the milk to a boil. Sprinkle in the semolina, whisking constantly over very low heat. Add the sugar, then simmer for 5 to 6 minutes, stirring from time to time. Remove from the heat and add the butter and the eggs, 1 by 1, stirring. Blend in the vanilla. The mixture will be thick but pourable, like a sauce.

Preheat the oven to 375 degrees F.

Butter a deep 10-inch baking pan. In the bottom of the pan, layer 8 sheets of phyllo, leaving the edges hanging over the sides of the pan. Brush each sheet generously with clarified butter. Pour the milk mixture over the phyllo layers.

Cut the remaining sheets of phyllo about the size of the pan and brush each sheet with clarified butter. Place the buttered sheets on top of the filling, then brush the hanging bottom sheets with clarified butter and roll them up to seal.

With a very sharp knife, score the top diagonally in 2 directions to make diamond-shaped slits on the surface. (This makes it easier to cut later, and also allows the syrup to penetrate the pie.) Sprinkle with a little water and bake about 1 hour, or until top is light brown and filling has set.

The minute you start baking the pie, begin to make the syrup. Simmer the water with sugar and the 1/2 of lemon for about 1 hour. Remove the lemon half and squeeze the juice into the syrup. Discard the lemon. Pour the syrup over the pie the minute you take it out of the oven. Be careful, syrup will bubble vigorously and can burn you.

Let cool, and serve, cutting pieces along the slits that you made before baking the pie