Chestnuts have their own unique magic, but can be a chore to peel in any serious quantity. We get around that by infusing just a few into the brown butter, which coats the whole dish with toasty chestnut flavor. Add a kiss of freshly grated nutmeg just before serving to amplify the holiday vibe.
— Shauna of SecretLanguagePDX.com
With a sharp paring knife, cut an X into the flat side of each chestnut. Arrange on a baking sheet cut side up and roast at 400 F until the skins begin to curl and detach from the nut, about 15 minutes.
Let cool *just* until you are able to handle them, and then use a paring knife to gently peel the skin away from the flesh. Discard the skins. Break or slice each peeled chestnut into small pieces and reserve.
Melt 2 tablespoons of butter in a medium sized saucepan over medium and add the rinsed leeks and a sprinkle of salt. Sweat the leeks, covered, stirring every few minutes, over low to medium heat until melty and tender, about 15 minutes.
In a medium sized pot, cover the potatoes in cold water and add a hefty pinch of salt. Bring to a boil over medium heat, then taste the water and adjust until it is quite salty (but not disagreeably so.) Reduce to a simmer, and cook until tender- a knife should easily slide in and out of the potato without resistance. Drain, let cool 5-10 minutes, and then using your fingers, break each potato into two or three pieces. Set aside in a mixing bowl while you quickly prepare the chestnut brown butter.
In a saucepan over medium-high heat, melt the remaining butter with the chestnut pieces. Whisk constantly until the butter foams, and then becomes golden brown (NOT deep brown). The chestnuts should continue to cook through and infuse their aroma into the butter without becoming too dark.
Remove from the heat and immediately add the cooked leeks. This will drop the temperature of the butter just enough so the browning stops at the perfect shade of toasty golden brown. Stir with a spatula to incorporate, making sure to work in the caramelized butter solids from the bottom of the saucepan. Drizzle the entire mixture over the warm broken potatoes. Season with flaky salt, fresh ground black pepper, and a squeeze of lemon, and garnish with parsley and nutmeg, if using.