Whisk the eggs along with half the sugar in a mixer for 3-4 minutes until fluffy and thick. Set aside.
In a large pot, place the heavy cream, milk, remaining sugar, a pinch of salt, 1 teaspoon vanilla extract and cook over medium heat to bring to a boil.
Slowly add the semolina, in a steady, thin stream, whisking vigorously. Whisk for about 3-4 minutes until it thickens. When the mixture peels away from the sides of the pot, it’s ready.
Turn off heat. Whisking constantly and vigorously, quickly pour in the egg-sugar mixture. Remove from heat and whisk in the butter.
Preheat oven to 350F.
Melt the butter in the microwave or in a saucepan. Generously brush a large baking pan with some of the melted butter.
Start assembling the pie by spreading a sheet of phyllo dough in the pan and drizzle and brush gently with butter. Repeat the same process with the remaining 5 sheets of phyllo dough. Very gently press down on the phyllo with your hands, just a little.
Spread the custard on top and turn in the phyllo that is hanging over the edges. Drizzle with butter. Place the remaining sheets over the custard, drizzling and brushing each one with melted butter. Fold in the edges, under the bottom sheets. To seal the edges of the galaktoboureko.
Drizzle a generous amount of melted butter over the top of the sweet and spray or sprinkle with a little cold water.
Score the top of the pie into 10-12 pieces and then cut into them and pour the remaining butter on top.
Bake for about one hour or until golden brown.
When ready, remove from oven and immediately pour the cold syrup over the pie.