Honeydew melon was in season when Kristina Burke and her husband, chef Jim Burke, opened their restaurant, so they decided to use it in a drink. The cucumber and rose flavors in Hendrick's gin balance the melon's almost unctuous sweetness.
In a cocktail shaker, muddle 4 of the honeydew balls with the gin. Add ice and shake well. Strain into an ice-filled rocks glass and stir in the chilled tonic water. Garnish with the remaining 3 honeydew balls and the thyme sprig.