1. In a mixer, beat the egg whites until stiff peaks form. Beat the yolks and slowly add them to the meringue, mixing gently. In a large saucepan, heat the milk and sugar until the sugar dissolves. Sprinkle in the semolina like rain, not all at once, and stir with a wooden spoon until it starts to bubble. Remove from the heat and add the butter and vanilla. Stir, and after 5 minutes, slowly add the eggs while stirring quickly.
2. Butter a 30x30x5 cm baking pan—it should be shallow so the syrup soaks in well. Place 6 phyllo sheets on the bottom, brushing butter between each layer. Pour in the cream and cover crosswise with the remaining 6 sheets. Use the 7th sheet to create a nest-like top layer. If any butter is left over, pour it over the top sheets and sprinkle with a little water.
3. Make 3 holes with a skewer and bake at 170-180 degrees Celsius for 45 minutes. Pour the syrup over and serve.