Combine the cranberries, diced apple, both sugars, apple cider vinegar, and water in a medium heavy-bottomed saucepan. Stir everything together and set the pan over medium heat. You will hear the liquid start to bubble within 3 to 4 minutes.

Once the mixture reaches a steady simmer, stir in the grated ginger, cinnamon, allspice, cloves, cayenne if using, and salt. The cranberries will start popping open as they heat up. This is exactly what you want. Keep the heat at medium so the mixture bubbles actively but does not boil so hard that it spatters or scorches on the bottom.

Cook uncovered for 20 to 25 minutes, stirring every 3 to 4 minutes and scraping the bottom of the pan each time. As the cranberries break down and the liquid reduces, the chutney will thicken noticeably. You will know it is getting close when the mixture coats the back of a spoon and a trail from your finger holds its shape for a couple of seconds.

At the 20-minute mark, taste carefully and adjust sweetness. If it is sharper than you like, stir in another tablespoon of brown sugar and cook for 2 more minutes. The apple pieces should be very soft but still visible in the mixture, giving the chutney its chunky character. If you prefer it smoother, press lightly with the back of your spoon to break down a few more apple pieces.

Remove the pan from heat and stir in the orange zest. The chutney will continue to thicken as it cools. It should be spreadable and jammy at room temperature, not runny. Transfer to a glass jar or airtight container. Let it cool to room temperature before sealing and refrigerating. Serve warm, at room temperature, or straight from the jar alongside the turkey platter.