Brown the bacon in the olive oil in a large skillet over medium heat. Remove the bacon from the pan and drain on a paper towel-lined plate. Place the romaine halves in the same pan, cut-side down, and brown in the rendered bacon fat, turning as needed until wilted, about 4 to 6 minutes. Remove the lettuce to the plate with the bacon. Add the wine to the skillet and use a wooden spoon to scrape up any browned bits that cling to the pan. Add the broth, thyme, and mustard and bring to a simmer. Return the lettuce to the skillet, and season with salt and pepper. Bring the liquid to a boil over high heat, and then adjust heat to maintain a gentle simmer. Place a circle of parchment paper directly on the lettuce, or set a lid slightly ajar on the skillet. Cook the romaine until tender, about 30 minutes.
Transfer the romaine to a serving dish and reduce the cooking liquid to a saucy glaze. Remove the thyme sprig and pour the cooking liquid over the lettuce, sprinkle with the bacon, and serve.
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