Preheat oven to 450 degrees F. Line a baking sheet with parchment paper or a silicone mat.
Roast peach halves for 15 to 20 minutes.
Add peaches to a blender and puree until smooth.
Add peach puree to a shaker filled with ice, along with gin, simple syrup, and lemon juice.
Strain into two cocktail glasses filled with ice. Top with club soda and garnish with a sprig of basil.