To make the syrup: In a small saucepan, combine the water, sugars, pumpkin purée, cinnamon, nutmeg, a small pinch of ground cloves and a pinch of kosher salt. Bring to a boil over high heat, reduce to a simmer and cook for 2 minutes, stirring to dissolve the sugar.
Add the vanilla extract and stir to combine. Remove the pot from the heat and let cool for 10 minutes. Strain the syrup into a Mason jar, seal and refrigerate until cool, 1 hour. The syrup can be stored in the refrigerator for up to 1 week.
To make one mule: In a cocktail shaker, combine the vodka, pumpkin spice simple syrup and lime juice. Fill the shaker halfway with ice and shake vigorously until frosty, about 10 seconds.
Fill a lowball glass halfway with ice. Strain the drink over the ice, top with ginger beer and garnish with a lime wedge.