Melt the butter in a saucepan over medium heat; cook the shallot in the melted butter until softened, about 5 minutes. Pour the vinegar, water, and 3 tablespoons of the bourbon into the saucepan; stir the mint and sugar into the liquid until the sugar is dissolved. Increase the heat to high and bring to a boil; immediately remove from heat and allow the mixture to steep for about 15 minutes. Strain through a fine mesh strainer. Stir 1 tablespoon bourbon into the sauce to finish.