Preheat the oven to 400 F. After 1 hour of chill time, remove the crust dough from the fridge. Use a rolling pin to roll out the dough into a rough circle, about 1/4" thick. Add the dijon mustard onto the dough and spread out evenly everywhere except for a 2 inch border on the edge.
Add parmesan cheese and evenly spread on top of the mustard. Add the caramelized onions on top of the cheese. Then, starting from the outside, layer the sweet potatoes, one at a time, slightly overlapping each other, in a circle until the entire galette is covered (other than the 2 inch border).
Fold the border of the crust over towards the middle. Rotate the galette and continue to fold the edges towards the center, until all the edges is folded in.
Crack the egg into a small bowl and whisk to combine. Use a pastry brush to brush the egg wash onto the crust. Transfer to a baking sheet and bake for 30-40 minutes, until the crust is golden brown. Remove from the oven, top with a drizzle of herb oil, slice, and serve immediately!