1. In a food processor, pulse the coffee beans until cracked, then transfer to a 1-liter, sealable jar. Pour in the Armagnac and seal shut. Let the jar sit on the counter, until infused, 2 days, shaking halfway through. Strain the Armagnac, discarding the beans.
2. Transfer the coffee-infused Armagnac to a medium saucepan with the sugar over medium heat. Gently cook, stirring constantly, until the sugar has dissolved, 4 to 5 minutes. Cool completely, then store in a sealable, 750-milliliter bottle in the refrigerator.