Whisk together eggs and cornstarch in a bowl and set aside. In a saucepan, heat fresh milk, sugar, and vanilla until it reaches a boil. Gradually whisk in the hot milk into the egg mixture, then return to heat and whisk until thickened, about 5 minutes. Set aside to cool.

Preheat your oven to 200°C (392°F). Grease a 15x10 inch baking dish with melted butter. Layer five phyllo sheets crumpled for added texture, drizzling melted butter between each layer.

Pour the cooled custard filling evenly over the crumbled phyllo base. Layer additional phyllo sheets on top in a similar crumpled style, drizzling with any remaining melted butter.

Place the bougatsa in the oven and bake for 10-15 minutes, or until the top is golden brown and crispy.

Allow to cool for a few minutes after baking. Dust with icing sugar and ground cinnamon before serving. Serve warm.