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Step 1
Make a brown sugar simple syrup: Add the brown sugar and 1 1/2 cups water to a small saucepan. Stir to combine, then heat over medium and cook until the sugar dissolves, about 5 minutes. Remove from heat and set aside. (Makes about 2 1/2 cups.)
Step 2
While the simple syrup is simmering, juice the limes into a large bowl until you have 2 cups lime juice and pulp. Add 1 3/4 cups of the simple syrup, plus the rum, bitters and nutmeg; stir to combine.
Step 3
Using a fine-mesh sieve, strain the punch into a large pitcher. Add 3 cups ice and stir until most, if not all, of the ice has melted and the pitcher is cool to the touch. (Taste the punch at this point and, if desired, add more simple syrup or lime juice.) If not serving right away, refrigerate up to 2 days.
Step 4
Serve in ice-filled rocks glasses. Garnish with grated nutmeg or a lime slice, or both.