Add all the ingredients into a shaker with ice and shake vigorously.
Double-strain into a chilled Martini glass.
Add a few drops of basil oil onto the foamed surface using an eyedropper, then crack one turn of black pepper over top.
*Honey syrup: Mix equal parts water and honey in a saucepan, and heat until melted. Cool and bottle.
**Basil oil: Plunge 1 lb basil leaves, with stems removed, into boiled water, and stir for about 10 seconds, then place in a bowl of iced water. Remove basil and gently press out excess water. Place basil in salad spinner to further dry the basil. Add the basil to a Vitamix and set on low. Add a small splash of vegetable oil and blend. Gradually increase speed while adding small amounts of oil until the Vitamix just starts to break down the basil. Turn the Vitamix to high, adding a little bit more oil, and continue to blend for 60 to 90 seconds. Use as little vegetable oil as possible (approximately 8 to 12 oz). Pour into a cheesecloth-lined chinois resting over a bowl and let gradually strain. Allow to cool, then bottle.