Cut puff pastry into 8 rounds with 4-inch ring mold. Brush pastry rounds with egg wash, sprinkle with Sugar in the Raw and bake until golden brown, about 8–10 minutes. Set aside to cool.
Heat large 10-inch sauté pan over medium heat. Add granulated sugar and cook until sugar reaches 330 degrees on candy thermometer (it will be light caramel in color). Try not to stir too often. Add butter, cinnamon stick, split vanilla bean, and salt, then add apple wedges.
Keep pan on heat; caramel will seize up once apples are added, so gently mix apples with rubber spatula until caramel becomes fluid again. Continue to cook until apples slightly soften and caramel sauce has reduced by half, about 5 minutes. As caramel starts to stiffen again, deglaze pan with cider and turn heat to high. Cook for an additional 3–4 minutes. Remove from heat. Note: apples should be soft and cooked through.
With slotted spoon, transfer apples to a sheet tray lined with parchment paper and allow to fully cool. Reserve caramel sauce in pan away from heat.
To assemble tart, arrange 8 cooled apple wedges in spiral on top of each precooked puff pastry round. Place rounds topped with the apples in 350-degree oven for 5 minutes. Garnish each tarte tatin with reserved cider caramel sauce and serve immediately. Serving with vanilla ice cream is optional, but advised.