Place the oven rack in the middle and preheat to 375° F/190° C.
In the large skillet, warm the olive oil over medium heat. Sauté the potatoes until they start to soften, about 3 or 4 minutes, then add the shallots, mushrooms, and ham and continue cooking until the all the vegetables are tender and browned, about 6 to 8 minutes. In the small bowl, whisk the eggs and green onions together, then pour into the skillet and add the goat cheese. Scramble the eggs with the vegetables, just until the eggs are set, not dry. Remove from heat and allow to cool.
Remove the puff pastry from the package and lay out on a floured surface. Roll into a rectangle 10 inches by 12 inches. Fold the puff pastry in half and transfer to the parchment on a cutting board. Using a sharp knife, make angled cuts all the way down the edges of the pastry evenly spaced approximately 1 inch apart. (This is approximate, as you can see below, mine aren’t perfect.) The center of the pastry should be about 4 inches wide. Trim the triangles from the top edges and the triangles from the bottom edges, leaving a short rectangle at the base about 4 inches wide. Transfer the parchment to the baking sheet.
Arrange the filling on the puff pastry in the middle section, leaving space at the top and bottom of the pastry for folding. Fold the top edge over the filling and begin braiding the strips, laying them across the middle. When you reach the bottom, fold it over the filling and tuck the last braids under, trimming away any extra. Brush the braid all over with the egg white wash.
Bake in the preheated oven until the pastry is crisp and golden, about 30 to 35 minutes. Remove from the oven and allow to cool about 5 minutes, then slice with a sharp knife to serve.