Heat the milk in a 3-quart / 2.8 L saucepan over medium heat until steaming but not boiling, 4 to 5 minutes.

Whisk the semolina into the hot milk gradually, stirring constantly to avoid lumps. Cook over medium-low heat for 5 to 6 minutes, stirring, until it thickens like a loose pudding.

In a separate bowl, whisk the eggs and sugar until pale. Temper by adding a spoonful of the hot semolina mixture at a time, whisking, until warmed through.

Pour the egg mixture back into the saucepan. Cook over low heat for 2 to 3 minutes, stirring constantly, until the custard coats the back of a spoon and no longer tastes of raw egg. Remove from heat and stir in the butter, lemon zest, vanilla, and salt. Set aside to cool for 5 minutes.

Heat the oven to 180 C / 350 F and butter a 33 x 23 cm / 9 x 13-inch baking dish.

Layer 5 phyllo sheets in the dish, brushing each sheet with melted butter before adding the next, letting the edges hang over the sides.

Pour the cooled custard over the phyllo base and spread it evenly with a spatula.

Cover with the remaining 5 phyllo sheets, brushing each with butter. Tuck or trim the overhanging edges. Score the top layer into 12 squares with a sharp knife, this makes cutting easier later and helps the syrup soak in.

Bake for 40 to 45 minutes, until the top is golden brown and the custard doesn't jiggle when you shake the pan gently.

While the pie bakes, combine the sugar, water, lemon juice, and lemon peel in a 1-quart / 1 L saucepan. Simmer over medium heat for 5 to 6 minutes, until slightly syrupy. Remove the peel.

As soon as the pie comes out of the oven, pour the hot syrup evenly over the hot pie. Let it sit for at least 60 minutes so the syrup soaks in before cutting.

Cut along the scored lines and serve warm or at room temperature.