Cut off dark green ends of leeks. Slice through white part lengthwise until you get to the root -- do not cut through the root. Clean leek layers under cold running water. Once the leeks are free of dirt or grit, cut through the root to make two long pieces of leek.
Use a sauté pan large enough to hold the leeks in one layer. Heat butter over med-high heat. When butter begins to foam, turn heat down to medium; add minced garlic and then the leeks, cut side down. Cook 1-2 min to brown them a little. Turn them over, sprinkle with salt, cook the 2nd side for 1-2 min.
Turn leeks back over so cut side is down, sprinkle with sugar, thyme and a pinch more salt. Add white wine (or stock) with bay leaf and bring to a gentle simmer. Cover and cook 35-45 min over med-low heat.
When leeks are tender enough that a knife pierces them easily, uncover pan and bring braising liquid to a boil. Let this reduce by half, then turn off the heat. Add the parsley, swirl it around and serve.