In a cocktail shaker, combine the cherry tomatoes, olive brine, Champagne vinegar, flaky sea salt, basil leaves, and lemon juice. Gently muddle until the tomatoes have released their juices and the basil is fragrant.
Add the gin and tomato vermouth, then fill the shaker with ice.
Shake vigorously until well chilled.
Double strain evenly into two chilled coupe or Nick & Nora glasses.
Garnish each cocktail with a cherry tomato and finish with a few drops of extra virgin olive oil just before serving.