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Cut off ends of leeks until you get to shank; a little light green part is fine, but not too much. Save unused portion of leeks in freezer for making stock later.
Slice leeks in half lengthwise until you get to root end—do not cut through root just yet. Clean leeks under cold running water. Once they are free of any dirt or grit, cut through root to make two long pieces of leek.
In large sauté pan, heat butter over medium-high heat. When it has melted and begins to foam, turn heat down to medium and add minced garlic, then leeks, cut side down.
Cook 1-2 minutes, to get them a little browned and to let butter get into leeks.
Turn over and sprinkle with sea salt, then cook other side 1-2 minutes.
Turn leeks back over so cut side is down, sprinkle with thyme leaves and a touch more sea salt. Add white wine and bring to gentle simmer. Cover and cook 35-45 minutes over medium-low heat.
When leeks are tender enough that a knife blade pierces them easily, uncover pot and bring braising liquid to rolling boil. Let this reduce by half, then turn off heat. Add parsley, swirl it around, and serve.