Start with the syrup. In a saucepan, add all the syrup ingredients and bring to a boil. Once it starts boiling, cook for 5 minutes. Let it cool completely.

For the custard: In a saucepan, add half the milk, the heavy cream, sugar, and the flavorings. Heat over medium heat.

In a bowl, combine the remaining milk, eggs, and cornstarch. Whisk until the cornstarch is fully dissolved.

Gradually add a little of the hot mixture from the saucepan into the bowl, whisking constantly, to warm up the egg mixture.

Pour the contents of the bowl back into the saucepan and whisk constantly until the custard thickens.

At this point, stir in the cold butter until melted, then cover the custard with plastic wrap and let it cool completely.

Once the custard is cool, melt the remaining butter. Cut the phyllo sheets in half, then cut each half lengthwise again to make 4 strips.

Take one strip of phyllo, brush it with butter, and layer another strip on top. Place a heaping tablespoon of custard on the bottom edge, fold the phyllo over once to cover the filling, then fold in the sides and roll up into a log.

Generously butter a large baking dish and arrange the rolls inside. If you have any leftover butter, pour it over the rolls.

Bake in a preheated convection oven at 300°F (150°C) for 40 minutes, then increase the temperature to 350°F (180°C) and bake for another 10 minutes.

As soon as you take the rolls out of the oven, pour the cold syrup over them and let them sit for a few hours to absorb the syrup well.