Slice the ham if not already sliced. In a mixing bowl, combine the cream cheese and Italian dried herbs and stir until smooth. Taste and season with salt and pepper.
Lightly dust a work surface with flour and roll out the puff pastry into a rectangle. Place the pastry so the long side is facing you.
Spread the herbed cream cheese evenly over half of the pastry. Layer ham slices over the cream cheese and sprinkle with about half of the grated cheese. Fold the uncovered side of the pastry over the filled side to enclose the filling.
Repeat the filling on the exposed half: spread more cream cheese, add remaining ham and cheese, then fold again so you have a layered pastry roll. Chill the roll in the refrigerator for at least 5 minutes to firm up.
While the roll chills, preheat the oven to 200°C (392°F) with top and bottom heat and line a baking sheet with parchment paper.
Use a sharp knife to cut the chilled roll into about 15 equal strips. Hold each strip by the ends and twist the ends in opposite directions to form a spiral. Arrange the spirals on the prepared baking sheet, leaving space for expansion.
Beat the egg with the tablespoon of milk in a small bowl. Lightly brush the tops of the spirals with the egg wash.
Bake for about 20 minutes or until the sticks are puffed and deep golden brown. Remove from the oven and let cool slightly before serving.
These can be made ahead and reheated briefly before serving. Store leftovers in an airtight container and re-crisp in a warm oven if desired.