Preheat oven to 375 degrees Fahrenheit (190 degrees Celsius). In a large frying pan over medium heat, add 1 tablespoon olive oil followed by the chopped onion. Saute until translucent, stirring occasionally.
Turn up heat to high and add coconut sugar to onions until they are golden brown. Remove from heat & pan, set aside.
Return pan to heat, add remaining olive oil, garlic, mushrooms, sea salt and pepper and saute until lightly wilted. Remove from heat and place on plate lined w/ paper towels. Fold paper towels around mushrooms and turn over, allowing them to drain. Transfer onions, mushrooms and sweet potato to freezer to cool for 10-15 minutes.
When the veggies are cool, unfold puff pastry on a lined baking sheet. Spread half of caramelized onions over middle third of pasty leaving about ¾ inch (2 cm) on each side. Follow with a layer of mashed sweet potato, then a layer of mushrooms and top with Dijon mustard and sprigs of rosemary. Season with sea salt and pepper, if needed.
Roll the pastry VERY CAREFULLY over the top of the mushroom mixture until it forms a log. Press down on the pastry edges to make a seal. Gently roll the log over so the seam is on the bottom.
To make the vegan egg wash, whisk all the ingredients together in a small bowl. Brush a thin veil of the wash over the top and sides of the pastry. Using a sharp knife, cut zig-zags over the top of the pastry. Place in freezer for 10 minutes, then repeat with another layer of vegan egg wash.
Bake for 40 minutes, depending upon your oven time. Your pastry is done when the crust is golden brown.