Preheat a grill to medium heat and lightly oil the grates. Place a lightly oiled peach half, cut side down on the grill and cook for 3-5 minutes. Remove from heat and allow to cool.
Once cooled, cube 1/4th of the peach, and slice the rest of the peach for garnish.
Add cubed peach, basil, a squeeze of lemon juice and brown sugar to a cocktail glass and gently muddle until the peach is broken down and brown sugar is dissolved. Add bourbon to the glass and stir.
Fill the glass with crushed ice and top with ginger beer. Garnish with grilled peach slices and basil.