Heat the milk with the ghee or butter in a saucepan until melted and combined.

Grease a baking dish with ghee or butter. Place a sheet of phyllo dough in the dish and brush it with the warm milk and ghee mixture. Repeat, layering and brushing until you use half the phyllo dough. Spread the cooked ground beef mixture in the middle.

Continue layering the remaining phyllo sheets, brushing each with the milk and ghee mixture, until all the dough is used.

Cut the assembled goulash into squares. Beat the egg with the remaining milk and ghee mixture, then pour it evenly over the top. Sprinkle dried thyme on top.