Step 1

Place a rack in center of oven; preheat to 400°. Roll out puff pastry on a lightly floured surface to a 12" square. (If you have 2 sheets of pastry, stack and gently roll out to adhere.) Using a 10" cast-iron skillet turned upside down as your guide, trace a circle with a sharp knife around circumference of skillet. Reserve scraps for another use. Prick pastry round all over with a fork. Roll dough onto rolling pin, then unfurl onto a rimmed baking sheet and chill until ready to use.

Step 2

Toss leeks, oil, kosher salt, and ½ tsp. pepper on another rimmed baking sheet. Arrange leeks cut sides down and roast until tender and lightly browned in spots, about 20 minutes. Let cool. Remove tough, leathery outer layer, then transfer leeks to a cutting board.

Step 3

Meanwhile, bring vinegar to a simmer in 10" cast-iron skillet over medium-high heat and cook, stirring occasionally, until slightly reduced, about 1 minute. Add butter, sugar, and 1 tsp. thyme and continue to cook, stirring occasionally, until butter has melted, about 2 minutes. Remove from heat and let cool slightly.

Step 4

Arrange leeks cut sides down and side by side in rows in vinegar mixture, trimming as needed so they all fit snugly in skillet. If there are any gaping holes, fill in with small pieces. You may have a leek or two left, but better to err on the side of having too many than not enough.

Step 5

Brush leeks with mustard. Sprinkle with 2 oz. Parmesan. Drape chilled pastry over leeks, tucking edges inside skillet with a spoon. Cut three 3"-long slits into pastry (cutting all the way through) to let steam escape—this will help the pastry cook more evenly. Bake tart until pastry is deeply golden brown and puffed, 35–45 minutes.

Step 6

Let tart cool slightly, about 5 minutes. Carefully place a large plate on top of skillet. Using towels, invert tart onto plate. If some of the leeks get left behind in skillet, use a butter knife or offset spatula to release and nestle them back into tart.

Step 7

Top tart with thyme. Season with sea salt and pepper.

Step 8

Do Ahead: Dough can be prepped 8 hours ahead. Cover with plastic and keep chilled.