Preheat oven to 375 degrees.
Roll out the sheet of puff pastry dough into a rectangle that is about 10 inches across and 16 inches long. Carefully transfer the dough onto a piece of parchment paper that is slightly larger than the dough. Trim back the four corners, by removing a 1 by 3 1/2″ rectangle from each corner, leaving a 3″ wide space in the middle. This will give you the tails to fold over and serve as your guide for cutting the strips.
Mix together the apricot preserves and water, then brush onto the pastry dough. Move any chunks of apricot to the center.
Cut 1″ wide strips up each side of the pastry dough, leaving about a 3″ gap in the middle to form the center of the braid. Aim for the same number and width of strips on each side.
Layer on half of the prosciutto, followed by the grated fontina, then the shaved gruyere, then the strips of shaved asparagus, and lastly another layer of prosciutto.
Fold up the tail at the end closest to you, then criss cross the strips in an alternating pattern.
Aim for the tip of the strip to sit close to the base of the strip across and one up. Move from right to left as you work your way up the braid. Once you reach the other end, be sure to tuck in the tail prior to wrapping the last two strips on each side. The last strips can be folded directly across or back towards the opposite end of the braid. Use the parchment paper to transfer the braid to a baking sheet. Beat together the egg and water to make the egg wash, then brush on top of the pastry braid.
Bake in the oven for 25 – 30 minutes or until the the pastry has puffed up and the top has turned a glossy golden brown. Serve warm.