Preheat the oven to 400°F. Place a rimmed sheet pan in the oven to preheat as well.
Place a chestnut flat-side-down on a cutting board. Use a serrated knife to cut a deep “x” on the rounded side of the nut. (I find using a serrated bread knife to be the safest and easiest way to cut the chestnuts.) Repeat with remaining nuts.
Bring a small saucepan of water to the boil. While the water is coming to the boil, place a kitchen towel inside a medium bowl.
When the water is boiling, place a large handful of chestnuts (about 1 cup) in the pot. Boil for 2 minutes. Remove the nuts with a slotted spoon or spider and transfer them to the heated sheet pan. The boiling allows water to get beneath the inner membrane and the hot oven creates steam that helps lift the membrane from the chestnut. Roast for 7 minutes.
Tip the nuts from the sheet pan into the towel-lined bowl and bring the corners of the towel together to keep the nuts warm. Return the sheet pan to the oven.
Note: You might want to wear latex or rubber gloves for this step if your fingers are sensitive.
Remove one of the nuts from the bowl, taking care to close the towel and keep the rest of the chestnuts warm. The corners of the “x” will have peeled back to reveal the skin beneath. Using both hands, pinch the four corners of the shell together to loosen and crack the skin beneath. Then peel the shell and skin from the chestnut. Repeat with remaining warm nuts then steam-roast another batch and peel. Continue until all the nuts have been peeled. You should have about 1 ¾ lbs. (800g) peeled nuts.
To make the jam:
Put a few small plates in the freezer so you can check the set of the jam later.
Place the peeled chestnuts in a large stockpot or Dutch oven. Add a pinch of salt and enough water to cover the nuts by 2 inches. Bring to the boil, then reduce the heat and simmer until the nuts are very soft but not falling apart, 40-55 minutes. The time will vary depending on the size of your chestnuts. When done, the chestnuts should feel starchy and you should be able to easily smash them to a paste with a fork.
Drain the nuts, reserving the cooking water. You should have roughly 2 ¼ lbs. (1kg) cooked chestnuts.
Place the nuts in a food processor (you may need to work in batches). Measure out ⅔ cup of the cooking water, and set aside for use later. Process the nuts to a very smooth paste, adding some of the remaining cooking water as necessary to get them moving. The chestnut puree will get thick and look sandy as you process it; just keep adding cooking water until it moves again. Let your food processor run for a minute or longer to make sure the paste is quite smooth. If you use all the remaining cooking water to get your machine moving and need a bit more, just use tap water. The amount of water needed will vary depending on how powerful your food processor is and on the size of the bowl. Set the chestnut puree aside.
Slice the vanilla bean in half lengthwise. Using a paring knife, scrape the seeds from the vanilla bean into the sugar. Rub the vanilla seeds into the sugar to distribute them evenly.
Rinse the pot you used to cook the chestnuts. Place the vanilla sugar, the scraped vanilla bean pod, lemon juice and the reserved ⅔ cup (200mL) cooking water in the pot. Cook over medium heat, stirring frequently, until the sugar has dissolved. Increase the heat to high and bring the sugar mixture to a boil. Continue to cook over high heat until the sugar mixture starts to foam. Add the chestnut puree and whisk well to combine, taking care not to splash any of the hot sugar syrup on yourself. The jam will darken as you cook it.
Cook at a rolling boil for 3 minutes then check for a set using the plates you placed in the freezer: place a teaspoon or so of the jam on the frozen plate and wait 10 seconds for it to cool. Rotate the plate. The jam should not run like a liquid, but have the consistency of loosely set jam. It will not wrinkle like jams with pectin to tell you it’s done, so just look for a consistency you like. If it is too loose for your taste, cook the jam for another 30-60 seconds and check for a set again. When the jam has reached your desired consistency, remove the pot from the heat and ladle the jam into the clean, sterilized jars. Process the jam or cool it and then store in the refrigerator or freezer. The jam will keep in the refrigerator for up to one month and in the freezer for up to a year.