Prepare grill and lightly grease grill grate with olive oil.
Season cut sides of peaches with nutmeg and salt and place seasoned side down on the grill. Grill 30 seconds then turn 90 degrees and cook 30 seconds more. Flip and turn grill to low. Cook until peaches are soft and caramelized. Remove from grill and let stand until cool enough to handle.
Scoop peach flesh into food processor and purée until smooth. Add coconut milk, sugar, extract and bourbon, if using, processing to combine. Place in refrigerator to chill. Pour into ice cream maker and freeze according to manufacturer’s directions. Makes about 3 cups.