Place the chestnuts in a large saucepot and fill with water. Season generously with salt and bring the pot to a boil. Add the potatoes and continue to boil until the potatoes are fork-tender, 20 to 25 minutes.
Drain the cooked potato-chestnut mixture and return to the pot. Add the hot cream and cold butter and stir vigorously until combined. Check the seasoning; add salt, if needed. If the mixture is a little stiff, add more cream and butter until it reaches the desired consistency. Place in a serving bowl and cover with foil until ready to serve.