Step 1Trim the roots from the leeks, keeping the bottom intact. Trim the green tops so that the remaining leek is about 4 inches long. Split the leek in half lengthwise. Run the leeks under cold water to clean and dislodge any dirt. Drain, place in a bowl and toss with one-fourth teaspoon salt.
Step 2Heat a large (12- to 14-inch) saute pan over medium heat then add the olive oil. Place the leeks in the pan cut-side down, in batches if necessary. Cook over medium heat until caramelized, about 12 to 15 minutes. Turn the leeks and cook for a few seconds on the other side. Remove the leeks from the pan and place, cut side up, in a shallow baking dish (about 9-by-13-inch).
Step 3In the same pan, lightly saute the sliced shallot over medium-high heat for about 1 minute. Add the thyme and the white wine and cook about 1 to 2 minutes, until the wine reduces slightly. Add the stock or water and bring the mixture to a quick boil. Remove from heat.
Step 4Pour the liquid over the leeks in the baking dish until the leeks are almost, but not quite covered, adding more stock or water if necessary. Place in a 400-degree oven and cook until the root ends of the leeks can be pierced with very little resistance by a knife, about 25 to 30 minutes.
Step 5While the leeks are braising, cut the potatoes into one-half inch slices. Place them in a medium saucepan and fill with cold water, covering the potatoes by 1 inch. Season the water with one-fourth teaspoon salt. Cook over high heat until the water boils and test for doneness by gently piercing with a knife. If necessary, reduce the heat to medium and cook for 2 or 3 minutes longer until done. Drain and set aside.
Step 6When the leeks are done, heat the saute pan over medium heat. Add the butter, and when it is melted, add the potato slices. Saute until lightly browned, about 1 to 2 minutes, turning as needed to cook both sides. Add the leeks and braising liquid to the pan and toss to combine. Season with salt and pepper to taste before serving.