For the buttermilk-rum caramel syrup: Combine the sugar, buttermilk, butter, corn syrup and baking soda in a medium saucepan and cook over medium-high heat, whisking constantly, until amber brown, 8 to 10 minutes. Remove from the heat and stir in the rum, vanilla and sea salt. Transfer to a heat-resistant bowl, cover and keep warm.
For the gingerbread pumpkin waffles: Preheat the oven to 250 degrees F. Place a baking rack on top of a baking sheet.
In a large bowl, whisk together the flour, granulated sugar, muscovado sugar, baking powder, baking soda, cinnamon, ginger, cloves, nutmeg and salt. In a medium bowl, whisk together the eggs, milk, butter, pumpkin puree, sour cream and molasses until smooth. Add the wet ingredients to the bowl with the dry ingredients and whisk until the mixture is just combined. Fold in the candied ginger. Cover and let sit at room temperature for 15 minutes.
Preheat a waffle iron and brush the grates liberally with butter. Fill the waffle wells and cook according to the manufacturer¿s instructions until crisp and golden. Transfer to the prepared baking sheet and keep warm in the oven until ready to serve. Repeat with more melted butter and the remaining batter. Serve with the buttermilk-rum caramel syrup.