Combine the milk, semolina, vanilla, 3 cups sugar, and eggs in a large saucepan. Cook over low heat, stirring constantly, until the custard thickens.

Remove from heat and stir in 2 tablespoons of butter.

Brush a large baking pan well with melted vegetable butter. Layer 6 sheets of phyllo dough in the pan, letting the edges hang over the sides. Place each sheet loosely, not pressing them down, and arrange so the overhanging edges are spread out around the pan. Brush each layer with melted vegetable butter as you go.

Pour the custard over the phyllo. Fold the overhanging phyllo edges over the custard, brushing each with a little more melted vegetable butter.

Layer the remaining 6 sheets of phyllo on top, brushing each with melted vegetable butter. Tuck in any overhanging edges with a knife. (If your pan is smaller than the phyllo sheets, trim the excess as needed. For a half batch in a smaller pan, use 4 sheets on the bottom, fold over the excess, and top with 3 more sheets.)

Lightly score the top with a sharp knife. Drizzle the remaining 2 tablespoons of melted butter over the top, then sprinkle with a little water to help the phyllo crisp up.

Bake in a preheated oven at 350°F (180°C) for about 1 1/2 hours, on the middle rack, until golden brown. If the top browns before the bottom, move the pan down one rack to finish baking the bottom layers.

As soon as you remove it from the oven, spoon the cooled syrup (made by boiling 3 cups sugar, 3 cups water, and a strip of lemon peel for 12 minutes, then letting it cool completely) evenly over the hot pastry.

Let the syrup soak in, then serve warm or cold.