Clean the mushrooms: Fill a large bowl with cold water, and have a clean, lint-free towel nearby. Add a few mushrooms to the water at a time, about as many as you can easily handle in your cupped hands, and swirl them around in the water to loosen any dirt. Immediately transfer the mushrooms to a towel, patting them dry and laying them out, caps up, to finish air-drying.

Bring a pot of water to a boil. Add the cleaned mushrooms to the pot of boiling water and cook for 5-6 minutes, or 10 minutes for wild mushrooms. Blanch in ice water.

Bring all the brine ingredients to boil. Place prepared mushrooms in a glass jar and pour the hot brine over the mushrooms to cover. Close the lid (it will vacuum-seal) and let cool slowly. When cool, place in the refrigerator.

Make the martinis: If using vodka, shake with ice. If using gin, stir with ice. Combine the vodka or gin, dry vermouth, and pickled mushroom liquid. Strain into 4 chilled martini glasses. Garnish with pickled mushrooms.