Preheat oven to 400°F with rack in lower third position. Line a rimmed baking sheet with parchment paper. Place potatoes and 2 tablespoons of the salt in a large saucepan; add water to cover by 1 inch. Bring to a boil over high; reduce heat to medium, and cook, undisturbed, until potatoes can be pierced using tip of a knife but are still firm, 2 to 3 minutes. Drain well; let stand until cool enough to handle, about 10 minutes.
Return saucepan to medium heat; add oil, and heat until shimmering. Stir in onion and remaining 1/4 teaspoon salt; cook, stirring occasionally, until translucent, about 3 minutes. Reduce heat to medium-low; cover and cook, stirring occasionally, until onion begins to soften and brown around edges, about 18 minutes. Uncover and increase heat to medium-high; stir in water, scraping up any browned bits from bottom of saucepan. Cook 2 minutes, stirring often, until onion is browned and caramelized. Let stand until cool, about 5 minutes.
Roll puff pastry on a lightly floured work surface to a 14- x 12-inch rectangle. Transfer to prepared baking sheet. Lightly score a border around perimeter of puff pastry using a paring knife, about 1 inch from edges. Prick pastry all over using a fork, leaving a 1-inch border around edge.
Stir together cream, thyme, Dijon mustard, nutmeg, and 1 cup of the Gruyère in a medium bowl until combined. Spread cheese mixture in an even layer inside scored border of pastry. Arrange potato slices in overlapping layers over cheese mixture inside border; top evenly with caramelized onions. Top with remaining 1/2 cup Gruyère.
Bake in preheated oven until pastry is puffed and deep golden brown, about 30 minutes. Transfer baking sheet to a wire rack, and let cool 5 minutes. Garnish with black pepper and additional thyme. Slice into 16 (3 1/4- x 3-inch) pieces; serve warm or at room temperature.
