Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper.
On a work surface, roll the puff pastry into a roughly 10 by 16-inch rectangle. With a sharp knife, trim uneven edges to make a perfect rectangle. Evenly cut off the outer 1 inch of each side of the rectangle in strips; put the puff sheet on the baking sheet. Dip your finger in water and run around the top edges of the rectangle and replace the removed strips of pastry along the edges of the sheet, pressing lightly to adhere. With a fork, pierce the interior of the tart shell to prevent rising; do not pierce the adhered edges. Bake until the outer edges have puffed and are golden in color; about 15 minutes. Set aside.
Heat 2 tablespoons olive oil in a large skillet over medium-low heat. Add onions and thyme sprigs and season well with salt and pepper, to taste. Cook, stirring occasionally, until the onions begin to brown. Add the stock, a tablespoon at a time, as the pan gets dry, scraping and stirring the brown bits that are stuck to the bottom oft the pan. When the onions are caramelized to a dark golden color, remove from the heat and discard the leafless thyme sprigs (the leaves fall off while cooking).
When ready to serve, preheat the oven to 350 degrees F. Evenly spread the caramelized onions on the cooked pastry shell and heat in the oven until warmed through, 5 to 7 minutes. Remove from oven, to a cutting board and cut into wedges. Arrange on a serving platter and garnish each wedge with a sprig of fresh thyme. Serve immediately and enjoy!