Heat oven to 400°. Beat the egg and water in a small bowl and set aside.
Prepare a baking sheet with parchment paper and set aside.
Lightly flour a clean work surface. Unfold the pastry sheet then lightly dust top with flour. Use a rolling pin to roll lightly so that the creases are flattened out. Do NOT make the square any larger when rolling.
Brush square of pastry with egg wash. Evenly spread the Boursin cheese over the square field. Next, sprinkle the minced herbs and the chives over the Boursin. Be sure you are covering evenly, and are getting as close to the edges as possible. Next, follow with the Parmigiano-Reggiano and the cheddar cheeses.
Starting at the side closest to you, roll pastry up like a jelly roll. Use your fingers to guide the fillings and keep them tucked into the roll as best possible.
With the seam side down and using a serrated bread knife, carefully cut the roll in half. Cut each half in half. Then, cut each section into 2 or 3 1/2-inch slices. Place slices onto the parchment-lined baking sheet. Brush tops with the remaining egg wash.
Bake 14 to 18 minutes or until light golden brown. Garnish with additional minced chives or green onions, and serve warm or at room temperature.