In a large saucepan, combine the brown sugar, cinnamon stick, and grated nutmeg. Set over medium-low heat. Keep an eye on the pan until the sugar starts to liquefy, roughly 5 minutes. Then use a metal spoon to stir so the sugar melts uniformly.
Once the sugar has completely melted, introduce the red wine. The sugar will solidify, but that's okay. Keep stirring, scraping the spoon along the pan's bottom to dislodge the hardened sugar.
After all the sugar has lifted from the pan's bottom, incorporate the orange juice, grapefruit juice, and orange slices into the saucepan. Maintain medium-low heat for another 20 minutes so the flavors intensify and the sugar dissolves.
Filter the mulled wine into a jug (optional: return the orange slices to the jug) and let it cool to room temperature. Once cooled, seal the jug's opening with plastic wrap and let it sit for 5 days before serving.