Gather the spices and preheat the oven to 350°F (180°C).
Lightly crush cardamom and peppercorns using a mortar and pestle or by pounding them slightly in a plastic bag.
Place cardamom, peppercorns, cloves, cinnamon, and coriander on a cookie sheet or in a baking pan, and toast until very aromatic (about five minutes).
Meanwhile, bring the sugar and water to a boil in a small casserole. Lower the heat and add the now-toasted spices, ginger, and vanilla.
Simmer gently for 3 minutes, and then turn off the heat and cover the casserole. Let this mixture sit for 5-10 minutes. The longer it infuses, the stronger it will be, so taste as you go.
Strain the syrup through a fine-meshed sieve and pour into a very clean jar with a tight-fitting lid.
To make the cocktail, pour vodka, lime juice, Chai syrup, and pumpkin purée into a glass and stir. Fill glass two-thirds of the way up with ice cubes. Top with the ginger beer and some crushed ice. Serve immediately. Santé et bonheur!