When the venison has been at room temperature for at least 2 hours, start a fire in your fireplace. Allow the fire to burn until the flames have died down and the glowing wood provides a steady, even heat.
Make the saltwater solution: Fill a medium saucepan with the water, add the kosher salt, and bring to a boil over high heat. Turn off the heat and allow the solution to cool. When it's cool, transfer it to a spray bottle.
Tie a 2-foot-long piece of butcher's twine securely around the bone at the end of the venison leg. Test it to make sure it will hold the leg's weight. Tie the other end of the twine to the fireplace hook so the venison is suspended and the bottom of the leg hangs about 8 to 12 inches above the fire. Place a large pan under the venison to catch the drippings.
The leg will begin to spin; give the heel a twist in the direction in which it is turning. Every 10 minutes or so, check to see if the venison is turning. If it's not, give it a gentle push so it spins.
After about 15 minutes, begin to baste with the saltwater solution every 5 to 10 minutes, spraying the venison leg all the way around.
Feed the fire from both sides (but not from the front), pushing the wood to the center as it burns. As the meat cooks, don't add any more wood, but let the flames die down so the embers are cooking the venison.
After 90 minutes, begin basting the venison with the mint pesto (recipe below). Set aside half of the pesto as a condiment to serve with the finished venison.
The venison will roast for between 1 and 2 hours for a 6-pound leg and 21⁄2 hours for a 12-pound leg. Check for doneness by inserting an instant-read thermometer in the thickest part of the leg. For medium-rare meat, the temperature should be 130 degrees.
Transfer the venison to a sheet pan, cover lightly with aluminum foil, and let it rest for at least 15 minutes.
Carve the leg. Spoon the carving juices over each slice and top with a generous portion of mint pesto. (If you have leftovers, the venison makes an incredible cold-cut sandwich.)
Bring about 2 quarts of water to a boil in a stockpot over high heat. While the water heats, set up an ice bath in a large bowl full of ice and water. When the water is boiling, blanch the mint for about 30 seconds. Use a slotted spoon to transfer the mint to the ice bath.
Blanch the parsley for 30 seconds, then transfer it to the ice bath.
Remove the herbs from the cold water and squeeze them in a ball to remove the excess moisture. Roughly chop them.
In a large, heavy-duty blender or a food processor, blend the herbs, toasted pine nuts, and garlic, pulsing just until combined. With the machine running, drizzle the olive oil in slowly.
Pour in the simple syrup. Salt and pepper to taste.