To Make the Infused Bourbon: Place the mushrooms in a quart-size jar and pour in the bourbon. Cover and set aside in a cool, dark place to infuse for 5 days. The flavor should be good and strong by then. Strain the bourbon back into the bottle. It will keep in the refrigerator for up to 3 months.
Make the Cocktail: Pour the mushroom-infused bourbon, Cynar, and bitter into a mixing glass with ice. Stir, then strain into a cordial glass.