Peel the pineapple, remove the core, and slice it into 1/2-inch thick rings. Pat the rings dry with paper towels—this is crucial for getting a good caramelized crust when grilling. Set aside on a plate until ready to cook. I find that preparing the pineapple first gives you time to focus on the sauce while it waits.
In a medium saucepan over medium heat, melt the unsalted butter with the dark brown sugar, stirring frequently until the sugar dissolves completely and the mixture becomes smooth, about 2-3 minutes. Add the heavy cream, spiced rum, ground cinnamon, and sea salt, then reduce heat to medium-low. Simmer gently for 5 minutes, stirring occasionally, until the sauce thickens slightly and deepens in color. The sauce should coat the back of a spoon when ready. Stir in the fresh lime juice and lime zest at the very end for brightness. I prefer to add the citrus at the last moment so the fresh flavor isn't lost to the heat.
Heat your grill to medium-high heat (around 400°F if using a gas grill, or wait until coals are glowing hot if charcoal). Place the pineapple rings directly on the grill grates and cook for about 5 minutes per side until golden-brown grill marks appear and the fruit softens slightly. The sugars in the pineapple will caramelize beautifully and concentrate the natural sweetness. Transfer to a serving platter as each batch finishes.
Arrange the warm grilled pineapple rings on serving plates or a large platter. Place a scoop of vanilla ice cream alongside or on top of each ring, allowing it to melt slightly into the warm fruit. Generously drizzle the rum caramel sauce from Step 2 over the pineapple and ice cream, and serve immediately while everything is still warm.