Using a vegetable peeler, zest the lemons and two limes. Place the zest in a pint-size Mason jar with the sugar. Set aside the zested limes. (Keep the zested lemons for another use.) Seal the jar and shake well to combine. Let it sit for at least 3 to 4 hours, or even overnight, shaking it occasionally. It allows the sugar to extract the aromatic oils from the peels.
Extract the juice from the peeled and unpeeled limes, then strain it to get approximately ¾ cup of lime juice. (Any extra juice can be saved for later use.) Pour this juice into the citrus-infused sugar mixture in the jar. Reseal the jar and shake it until the sugar has completely dissolved. This mixture is your shrub.
Fill a 1-gallon punch bowl or pitcher halfway with ice cubes. Pour the shrub, including the citrus peels, into the bowl. Add the Cognac, maraschino liqueur, and soda water. Stir everything together thoroughly. Grate about ¼ of the nutmeg on top of the punch and let it sit for 15 minutes. Serve the punch in 3- to 4-ounce portions.