Put the yogurt in a paper towel-lined strainer set over a bowl, set aside to drain and thickened, about 60 minutes. Discard the watery liquid. Whisk the horseradish into the thickened yogurt.
Meanwhile, heat 2 tablespoons of the olive oil in a medium skillet, over medium heat. Add the onions, 1 thyme sprig, and season with salt and pepper, to taste. Cook, covered, until the onions have wilted, about 10 to 15 minutes, (give them a stir every now and then with a wooden spoon to prevent them from sticking). Add the red wine, and simmer over high heat until most of the wine gets absorbed into the onions. Add the honey and red wine vinegar and simmer gently until the onions get jam-like, about, about 15 minutes.
Preheat a stove top grill pan over medium-high heat. Whisk the remaining 2 tablespoons oil, garlic, and balsamic vinegar in a bowl. Strip and add the leaves from the remaining sprig of thyme. Coat the mushroom caps all over with the flavored olive oil. Grill the mushrooms, turning as needed, until tender but not mushy about 3 to 4 minutes per side. Season with the salt and pepper.
Sandwich the mushroom between the buns, top with some of the onion jam, some lettuce, and a dollop of the horseradish cream. Serve warm.
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