Beat the butter in a small bowl with a rubber spatula until creamy. Add the parsley, 1/4 teaspoon salt and the nutmeg and mix to combine. Scrape the butter onto a piece of plastic wrap and form into a log. Wrap tightly and refrigerate until firm, at least 30 minutes, or freeze up to 1 week.
Combine the milk, 1 teaspoon salt and the bay leaf in a large saucepan. Peel the potatoes and cut into 1-inch pieces, adding them to the pot as you go so they don't discolor. Bring to a gentle simmer over medium heat; cook, stirring to prevent the milk from scorching, until the potatoes are very tender, 15 to 20 minutes.
Drain the potatoes, reserving the milk in a liquid measuring cup or bowl; discard the bay leaf. Return the potatoes to the saucepan and add all but about 1 tablespoon of the herb butter. Mash with a potato masher, gradually adding the reserved milk as needed, until creamy (you may have leftover milk). Season with salt. Transfer to a serving dish and top with the remaining 1 tablespoon herb butter just before serving.