Heat the milk in a saucepan. Once warm, add the semolina and stir constantly until it thickens into a cream. Add 1 tablespoon of the butter and mix in (reserve the remaining 3 tablespoons for the phyllo and pan).

In a separate bowl, whisk together the eggs, sugar, and vanilla.

Once the semolina mixture has thickened, remove from heat. Add the egg mixture and stir well to combine.

Melt the remaining butter and brush it onto a baking pan. Layer the phyllo sheets in the pan, brushing each sheet with melted butter. Pour the cream filling in the center. Continue layering the remaining phyllo sheets on top, brushing each with butter, until all are used.

Preheat the oven to 350°F (180°C). Place the galaktoboureko in the oven without cutting it. Bake for 50 minutes.

Once baked, set aside to cool completely before adding the syrup.

To make the syrup, boil the water and sugar together for 5 minutes. Pour the hot syrup over the cooled galaktoboureko. Let it rest for 2-3 hours before serving.