1. In a food processor, process butter, parsley, shallot, lemon juice, 1 tsp. salt and pepper until smooth, stopping to scrape bowl as needed. Transfer to a piece of plastic wrap, shape into a log; roll up. If not using right away, chill.
2. Place potatoes in a large pot and add enough water to cover them by 1 inch. Season with a large pinch of salt and bring to a boil over high heat. Reduce heat to medium-high and cook until tender, 12 to 15 minutes.
3. Drain potatoes thoroughly and transfer to a large bowl. Add all of the parsley butter (if cold, chop it first) and 1 Tbsp. water. Toss until potatoes are evenly coated and glossy. Serve warm.