Make the pineapple simple syrup: In a medium heatproof bowl ,add the sugar. Pour the boiling water over the sugar and stir until the sugar dissolves. Set aside to cool completely, then add the pineapple chunks and cover the bowl with plastic wrap. Refrigerate overnight.
The next day, strain the pineapple syrup through a fine mesh sieve and discard the pineapple or reserve for another use. Transfer the syrup to a covered container and refrigerate until ready to use.
Make the cocktail: Fill a coupe glass with ice cubes and water. In a cocktail shaker filled with ice cubes, add the rum, lime juice, Chartreuse and ¼ ounce of the prepared pineapple syrup. Cover and shake until the liquid is chilled, about 10 seconds. Discard the ice and water from the coupe glass and double-strain the cocktail through a fine-mesh strainer into the chilled coupe. Serve immediately.