Drizzle chocolate syrup, forming a swirling design, inside a martini or coupe glass. Freeze until ready to use.

In a cocktail shaker, combine Baileys, bourbon, chocolate liqueur, Kahlúa, and milk.

Fill shaker with ice, cover, and vigorously shake until outside of shaker is very frosty, about 20 seconds.

Strain cocktail into prepared glass. Grate nutmeg over cocktail and garnish with an Oreo turkey (if using).