1. Chill a martini glass by filling it with ice and letting it sit while you prepare the cocktail.
2. In a mixing glass, add the gin or vodka, dry vermouth, and olive brine.
3. Fill the mixing glass with ice, and then stir the mixture well. This helps to properly dilute and chill the cocktail.
4. Discard the ice from the martini glass, and strain the cocktail into the chilled glass.
5. Drizzle the extra virgin olive oil on top of the cocktail. The oil will float on the surface, creating a beautiful presentation.
6. Garnish your Olive Oil Martini with a lemon twist or a few green olives on a cocktail pick.
7. Serve and enjoy your unique twist on the classic martini!

The olive oil in this cocktail adds a smooth, silky texture and a subtle, fruity flavor that complements the brininess of the olives. Make sure to use high-quality olive oil for the best results. You can also adjust the amount of olive oil to your taste, but be mindful not to overdo it, as it can quickly overpower the drink.