Place a rack in middle of oven; preheat to 400°.

Unfold 1 sheet frozen puff pastry on a lightly floured surface and roll out to ⅛"–¼" thick.

Using a 12"-diameter ovenproof stainless-steel skillet turned upside down as a guide, cut pastry into a 12" round; discard scraps.

Turn skillet right side up and coat bottom and sides with 3 Tbsp. unsalted butter.

Sprinkle pan with 2 Tbsp. brown sugar in an even layer; season with kosher salt.

Stir ⅓ cup harissa and 1 tsp. ground cumin in a small bowl to combine.

Arrange a layer of sweet potatoes in skillet; brush with harissa mixture. Repeat with remaining sweet potatoes and harissa mixture to make 2 more layers.

Drape pastry over sweet potatoes and tuck edges in between skillet and sweet potatoes.

Using a paring knife, cut about six 1" slits all over pastry to let steam vent while baking.

Place skillet over medium-high heat and cook just until sugar and butter start to melt around edges, about 1 minute.

Transfer skillet to oven and bake until pastry is golden brown and crisp, 30–35 minutes.

Immediately invert tart onto a large plate.

While the tart is baking, place sliced red onion in a small heatproof bowl.

Bring ½ cup white wine vinegar, 1 Tbsp. coriander seeds, and remaining 2 tsp. brown sugar to a boil in a small saucepan, then pour over red onion.

Season generously with kosher salt and stir to combine. Let cool.

Sprinkle tarte Tatin with flaky sea salt and cut into wedges. Serve with drained pickled red onion.