If the barrel is new and dry inside, fill it with water and let stand until watertight, about 24 hours. Drain.
Using a funnel, fill the barrel with all of the liquid ingredients. Let age, tasting a sample once a week, until the cocktail has taken on a rounded but not overly oaky flavor, about 1 month.
Strain the cocktail through a coffee filter–lined funnel into a glass container and store indefinitely.
To serve, pour 3 ounces of the cocktail into an ice-filled rocks glass and stir well, then garnish with a twist.