Roll crust and fit into a 4x13 inch tart pan with removable bottom, pressing the dough into the fluting. Prick the bottom of the crust every few inches with a fork and place crust in a freezer for 30-45 minutes.
Preheat oven to 350°F.
Remove crust from freezer, line with parchment, and add pie weights. Bake until crust looks dry on the bottom, about 20-25 minutes. Remove parchment and weights and continue baking until golden brown, about another 20 minutes. Remove and let cool.
While the crust is baking, melt butter in a large skillet over medium heat. Add onions, sprinkle with about 1/4 teaspoon salt and 1/8 teaspoon black pepper, stir and cook until beginning to soften, 5 minutes. Cover and continue to cook, stirring every 5 minutes, until onions start to turn golden, 20-25 minutes. Remove from heat and let cool.
In a medium bowl, whisk together 3/4 tsp salt, 3/8 tsp black pepper. goat cheese, eggs, thyme, and nutmeg until smooth. Stir in cooled onions.
Scoop mixture into tart shell (any extra can be baked in buttered ramekins or custard cups) and sprinkle with Parmesan cheese.
Bake until puffed and golden brown, about 45 minutes.
Cool 5-10 minutes before lifting from pan and placing on a rectangular plate.