To prepare the puddings, preheat the oven to 350 degrees F. Brush the inside of six 6-ounce heat-proof custard cups or ramekins with the melted butter. Invert the custard cups on a baking sheet and refrigerate until the butter sets.
In a medium bowl, whisk together the flour, baking soda, salt, cinnamon, nutmeg, and coriander.
In the bowl of a stand mixer fitted with the paddle attachment (or in a large bowl with a handheld mixer), beat the 8 tablespoons butter and the cane sugar on high speed until the mixture is light and fluffy. Add the eggs one at a time, followed by the yolk and the vanilla, beating well and scraping down the side of the bowl with a rubber spatula after each addition. Beat in the pumpkin puree and scrape down the bowl.
Add the flour mixture in thirds, alternating with the buttermilk, beating each time only until the batter is smooth.
Use an ice cream scoop or large spoon to carefully divide the batter among the custard cups, taking care to keep the rims clean. Gently tap each cup to remove any air bubbles.
In a lidded baking dish or Dutch oven large enough to hold the custard cups with at least 1/2 inch of space between them, place a folded kitchen towel (to hold the cups steady). Arrange the custard cups in the dish. Pour enough very hot tap water into the baking dish to come halfway up the sides of the cups. Cover the baking dish tightly with the lid.
Bake until a toothpick inserted into the center of a pudding comes out clean and the top springs back when gently pressed, 30 to 40 minutes. Remove the lid and let the puddings cool in the water bath for 20 minutes.
Meanwhile, prepare the hard sauce. In a medium bowl, beat the butter with an electric mixer set to high speed until fluffy. Reduce the speed to low and add the confectioners' sugar. When the sugar is incorporated, increase the speed to high, and beat until the mixture is light and fluffy.
Fold in the nutmeg and rum with a rubber spatula; you should have about 1/2 cup. Use within 1 hour or transfer to a bowl, cover, and refrigerate for up to 1 week. Allow the sauce to come to room temperature before serving.
The puddings can be served warm or at room temperature. If serving directly in the custard cups, serve at once or let cool on a wire rack. If unmolding, to serve the puddings warm as soon as they come out of the water, run a thin knife around the inside of each ramekin to loosen the puddings, then invert into a serving plate. To serve at room temperature, place the ramekins on a wire rack until the puddings cool before inverting them onto serving plates. Top each serving with about 1 tablespoon of the hard sauce and a large shard of brittle.
Pumpkin Seed Brittle:
Generously butter a rimmed baking sheet or line it with a silicone baking mat. In a large skillet, cook the pumpkin seeds over medium heat, stirring often, until they are lightly toasted, about 3 minutes. Transfer to a plate to cool.
In a large, heavy, deep saucepan, combine the cane sugar, corn syrup, 1/4 cup water, and 2 tablespoons butter. Cook over medium heat, stirring with a long-handled wooden spoon until the sugar dissolves. Increase the heat to medium high, and bring the mixture to a boil. Let the mixture boil without stirring until it turns deep amber, 8 to 12 minutes.
Remove the pan from the heat and carefully stir in the baking soda, vanilla, and salt. The mixture will bubble vigorously and expand in the pan. Immediately stir in the pumpkin seeds and pour onto the prepared baking sheet. Use a heat-proof spatula to gently but quickly spread the brittle into a thin, even layer.
Let the brittle cool to room temperature. Run an offset spatula under the brittle to help loosen it, or gently twist the pan. Break the brittle into shards. Store in an airtight container at room temperature for up to 2 weeks. Yield: about 1 pound.
Professional Recipe: This recipe was provided by a chef, restaurant or culinary professional and makes a large quantity. The Food Network Kitchens chefs have not tested this recipe in the proportions indicated and therefore cannot make any representation as to the results.