Skewer the orange slices and the cherries, and brush on both sides with vegetable oil. Grill until charred on both sides, 2 to 3 minutes total. Remove from the grill and let cool slightly.
Add the grilled fruit to a pitcher, reserving a couple orange rinds. Muddle the fruit with the turbinado sugar. Add the bitters and then pour in the bottle of bourbon. Stir to combine. Taste and add more bitters as desired.
To serve, rub the rim of a rocks glass with the reserved grilled orange rind. Add ice and then fill by two-thirds with the bourbon mixture. Top off with club soda and stir to combine.