Place the milk, eggs, butter, semolina, sugar, and vanilla into a large saucepan.
Cook over medium heat, stirring constantly, until the mixture thickens and comes to a gentle boil.
Remove from the heat and allow the custard to cool slightly.
Preheat the oven to 180°C.
Lightly butter a large baking dish.
Layer half of the filo sheets into the dish, brushing each sheet generously with melted butter.
Pour the custard evenly over the pastry.
Cover with the remaining filo sheets, again brushing each layer with butter.
Butter the top layer well and score the surface into serving portions with a sharp knife.
Bake for 50–60 minutes, or until the pastry is deeply golden and crisp.
Remove from the oven and allow the pie to cool completely.
While the pie cools, combine the water, sugar, lemon peel, and lemon juice in a saucepan.
Bring to a boil and simmer for 8–10 minutes until slightly thickened.
Remove the lemon peel and keep the syrup hot.
Slowly pour the hot syrup evenly over the completely cooled Galaktoboureko.
Leave to stand for several hours, or preferably overnight, allowing the syrup to absorb fully.
Serve at room temperature with a cup of Greek coffee or tea.
