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Heat oven to 150 degrees. On stovetop, heat cast-iron skillet to very hot. Brush steaks with 1/2 tablespoon oil. Season with salt and pepper. Sear steaks in skillet very briefly, less than a minute a side, just enough to brown. Place meat in a baking dish in oven for 1 1/2 hours.
For chimichurri, combine parsley, thyme, garlic and shallot in food processor and pulse until minced. Transfer to a bowl, add vinegar, paprika and 5 tablespoons oil. Season with salt and pepper. Set aside. Heat remaining oil in skillet used for meat, add mushrooms and sauté until wilted. Season with salt. Set aside.
When steaks are done, slice and place on a platter. Reheat mushrooms, remove from heat, toss with chimichurri and arrange alongside steak.