Preparation

Make the sauce

• Halve and core the unpeeled apple. Cut half into 1/2-inch pieces and half into 1/4-inch dice.
• Combine the sugar, corn syrup, and 1/2 cup of water in a 3-quart saucepan. Scrape the seeds from the vanilla bean into the pan and then drop in the bean. Stir until the sugar is wet and sandy. Cook over medium-high heat without stirring until the sugar begins to turn golden around the edges, about 8 minutes. Stir in the 1/2-inch pieces of apple and cook, swirling the pan occasionally, until the sugar is dark golden, 3 to 5 minutes more. Remove from the heat; slowly stir in the cider and cider vinegar, followed by the salt.
• Strain the sauce through a fine-mesh sieve into a small heatproof bowl, pressing as much of the apple pulp through the sieve as possible. Quick-chill the sauce by putting the bowl in a larger bowl of ice water and stirring occasionally until cool. Add the diced apple and refrigerate, covered, until ready to serve. (The sauce will keep for 2 days.)

Make the panna cotta

• In a small bowl, whisk the gelatin with 1/4 cup cold water and set aside.
• Whisk the cream, milk, brown sugar, salt, ginger, vanilla, cinnamon, and nutmeg in a 3-quart saucepan over medium heat until just beginning to boil.
• Remove the pan from the heat and whisk in the softened gelatin. Whisk in the pumpkin, and then strain through a fine-mesh sieve into a bowl or 2-quart liquid measure. Divide among six 6-oz. ramekins. Refrigerate until firm to the touch, covering with plastic wrap once cool, at least 4 and up to 48 hours.
• Unmold the panna cotta by dipping the bottom of each ramekin in hot water for a few seconds and then running a paring knife around the edge all the way to the bottom to loosen it. Cover with a dessert plate and invert, give it a good downward shake, and then remove the ramekin. Serve topped with the sauce.