Prepare the syrup: In a small saucepan, combine sugar, water, cinnamon stick, and the squeezed lemon. Bring to a boil and simmer for 6-8 minutes. Remove from heat, discard the lemon, and set aside to cool.
Prepare the custard: Set aside 1 cup (250ml) of milk and ½ cup (100g) sugar. In a large pot, heat the remaining milk, sugar, lemon zest, vanilla extract, and cinnamon stick over medium heat.
In a separate bowl, beat eggs and yolks for 5 minutes. Add the remaining sugar, beating until dissolved. Gradually mix in semolina, followed by the reserved 1 cup of milk.
Slowly add the warm milk mixture to the egg mixture, stirring constantly. Return to the pot and cook, stirring continuously, until small bubbles form at the edges (about 15 minutes). Remove from heat and discard the cinnamon stick.
Spread cold butter on the surface of the custard to prevent a crust from forming. Set aside to cool.
Prepare the phyllo: Melt butter and let it cool slightly. Grease a 13-15 inch round or 15×11 inch rectangular baking dish. Layer 7 sheets of phyllo at the bottom of the dish, brushing each with melted butter. Let the edges hang over.
Assemble the pie: Pour the cooled custard over the phyllo and spread evenly. Fold over the overhanging phyllo. Layer 5 more sheets of phyllo on top, buttering each sheet.
Score the top layer into pieces without cutting through the bottom layers. Sprinkle with water. Bake at 356°F (180°C) for about 1 hour, until golden brown.
Remove from oven and slowly pour the cooled syrup over the pie. Allow the pie to sit for at least 1 hour before serving to absorb the syrup.