In a small saucepan over medium-high heat, bring ginger, cinnamon, cloves, and 1 1/2 cups water to a boil. Reduce heat to medium and cook, stirring occasionally, until liquid is reduced by about half, about 20 minutes.

Add sugar and stir until dissolved. Let cool about 10 minutes. Strain sugar mixture through a fine-mesh sieve into a small bowl.

Place crushed cookies on a small shallow plate. Pour some gingerbread syrup onto another shallow plate. Dip rim of a martini glass into gingerbread syrup, then dip into cookies to coat.

In a cocktail shaker filled with ice, combine ice cream, Baileys, vodka, Kahlúa, and 1/2 oz. gingerbread syrup. Cover and vigorously shake until outside of shaker is very frosty, about 20 seconds.

Strain into prepared glass. Garnish with a generous dollop of whipped cream and sprinkle with cinnamon. Gently place a gingerbread person in cocktail.