In a Dutch oven or heavy-bottomed pan, stir together the carrots, sugar, lemon zest, lemon juice, and salt. Tie up the whole spices in a piece of cheesecloth and add to the carrots. (You can put the spices in loose, but you will need to fish them out before canning, so if you do, count those cloves!) Cover and let rest in the fridge to macerate overnight.
The next day, add the water and nutmeg to the carrot mixture. Bring to a boil over medium heat and boil the jam, stirring often to prevent scorching, until the syrup has mostly evaporated and is thickened and orange, and the carrots are glossy, 20 to 30 minutes. If you boil the carrots too long they will become gluey. It’s not the end of the world. Go ahead and process them, but loosen them up a bit with some warm water before serving.
Have ready 2 clean half-pint jars and bands, and new lids that have been simmered in hot water to soften the rubberized flange. Remove the spices, then pour the carrot mixture into the jars leaving ¼ inch of headroom. Wipe the rims, place on the lids, and screw on the bands fingertip tight.
Process the jars in a water bath for 10 minutes at sea level. Process the jars for an additional 1 minute for every 1,000 feet altitude. Remove from the water, let the jars rest for 24 hours, and then check the seals. Store in a cool, dark place for up to a year.