Spiced Fig Tarte Tatin

Name: Spiced Fig Tarte Tatin
Ingredients: 1 sheet frozen puff pastry, thawed (half a 17-ounce package); All-purpose flour, for sprinkling; 2 teaspoons cornstarch; 1 teaspoon lemon juice; 1 teaspoon pure vanilla extract; 3/4 cup sugar; 2 tablespoons unsalted butter, cubed, at room temperature; 1/2 teaspoon kosher salt; 1/2 teaspoon ground cardamom; 1/4 teaspoon ground cinnamon; 14 ounces fresh figs (about 20 medium figs), stems trimmed and figs halved lengthwise; Vanilla ice cream or whipped cream, for serving