Pickled Mushroom Martini

Ingredients

• 2 lbs. fresh mushrooms, such as cremini, oyster, and enoki
• 2 ¼ cups white vinegar
• 2 teaspoons whole mustard seeds
• 1 ½ teaspoons brown sugar
• 2 teaspoons kosher salt
• 1 teaspoon whole allspice
• 5 dried bay leaves
• 1 carrot, peeled and cut into ¼” inch slices
• 6 oz. vodka or gin
• 4 oz. dry vermouth
• 2 oz. mushroom pickle liquid
• Pickled mushrooms, for garnish