Using a chef's knife, trim the unpeeled potatoes into rectangular blocks. Cut each into 1/4-inch thick slices. Stack two or more slices together and cut into 1/4-inch julienne strips. Cut these strips into 1/4-inch dice and place in a saucepan, covered with cold water, until ready to cook.
Place the potatoes in cold water, bring to a boil, then immediately take off the heat and drain. Allow the potatoes to air dry. Meanwhile, prepare the persillade by mixing together the garlic and parsley. It is possible to prepare the dish ahead to this point.
If the potatoes aren't thoroughly dry, blot them with towels. Heat the oil in a very large skillet over medium heat. Add the potatoes and cook, tossing occasionally, until they brown lightly, 6 to 8 minutes. You may drain and discard oil at this point, if desired. Add the butter and continue sauteing for another 5 minutes or longer to brown the potatoes completely. Just before serving, toss the potatoes with the garlic and parsley over medium heat for about 1 minute. Season with salt and pepper. Serve immediately.