For the Liquid Olive:
Combine and blend olives, lemon juice, gin, brine, calcium lactate, and xantham gum. Slowly drop mixture in spheres into sodium alginate bath and let stand for 4 minutes. Remove from bath and rinse with water.
Bring water to just below a boil. Add Laphroaig and sugar and stir until sugar dissolves. Remove from the heat and use a smoking gun or vaporizer to add cherry wood smoke to the mixture. Seal with plastic wrap and let stand for 15 minutes. Transfer to a siphon.
To Assemble and Serve:
Combine gin and vermouth and stir. Strain into a chilled cocktail glass. Garnish with liquid cocktail olive and top with Laphroaig foam.