Preheat oven to 425°F. Cook bacon in a 12-inch stainless steel skillet over medium, stirring occasionally, until rendered and crisp, 10 to 12 minutes. Transfer bacon to a paper towel–lined plate using a slotted spoon. Reserve 5 tablespoons drippings in a small bowl; discard remaining drippings, or reserve for another use. Clean skillet.

Toss together leeks, potatoes, salt, 3 tablespoons reserved bacon drippings, and ¼ teaspoon pepper on a large rimmed baking sheet. Arrange vegetables in a single layer, with leeks cut sides down, and roast in preheated oven until tender and bottom sides are browned, about 20 minutes. Set aside. Leave oven on.

Meanwhile, whisk together vinegar, sugar, 1 teaspoon mustard, remaining ¼ teaspoon pepper, and remaining 2 tablespoons reserved bacon drippings in skillet. Cook over medium, swirling occasionally, until mixture is reduced to a thick syrup that just coats the bottom of the skillet, 5 to 7 minutes. Remove from heat; set aside.

Roll out puff pastry on a lightly floured surface into a 14- x 12-inch rectangle. Cut out a 12-inch circle using a pot lid or plate as a guide; discard scraps. Transfer pastry round to a parchment paper–lined baking sheet; spread evenly with remaining 1½ tablespoons mustard. Score dough all over with ½-inch-long slits. Chill in refrigerator until ready to use.

Transfer leeks to a cutting board, and slice crosswise into about 2-inch-long segments. Arrange potatoes and leeks in a tightly packed single layer, browned sides down, in syrup in skillet. Scatter ½ cup reserved cooked bacon over top, tucking bacon pieces into any gaps between leeks and potatoes. Sprinkle evenly with Gruyère. Arrange chilled pastry round, mustard side down, over cheese. Fold edges of pastry inward, as needed, to fit inside skillet.

Bake tarte Tatin at 425°F until pastry is puffed and golden, about 20 minutes. With tarte Tatin still in oven, reduce oven temperature to 375°F; bake until pastry is deeply browned, about 15 minutes. Let tarte Tatin cool in skillet for 5 minutes. Loosen pastry edges from skillet using an offset spatula, if needed. Set a wire rack inside a rimmed baking sheet, and carefully invert tarte Tatin. Let cool until no longer steaming, about 15 minutes.

Transfer tarte Tatin to serving plate or cutting board. Garnish with chives, flaky sea salt, and remaining cooked bacon. Serve warm.