1. Pierce the steak all over with a fork and place in a resealable plastic bag. Whisk 1/2 cup of the beef broth with 2 tablespoons of the Worcestershire, the tomato paste and balsamic vinegar in a small bowl until smooth. Pour over the beef and add the thyme sprigs and smashed garlic. Seal the bag and press the meat so it is covered by the marinade. Marinate, refrigerated, 4 to 6 hours.
2. Preheat the broiler. Line the bottom of a broiler pan or rimmed baking sheet with heavy-duty aluminum foil. If using a baking sheet, place a rack over the foil. Remove the meat from the marinade and discard. Pat the meat dry and sprinkle with 1 teaspoon salt and 1/2 teaspoon pepper. Pour the remaining 1 1/2 cups beef broth into the bottom of the pan and place the meat on top of the broiler pan or rack. Broil about 6 minutes per side for medium-rare, or until a thermometer inserted in the center registers about 125 degrees F. Let the meat rest on the rack for 5 minutes, and then transfer to a cutting board and let rest an additional 5 minutes. Reserve the juices in the bottom of the broiler pan or baking sheet.
3. While the meat rests, heat 1 tablespoon of the butter in a medium skillet over medium heat. Add the onions, chopped thyme and 1/4 teaspoon salt and cook, stirring, until the onions are very soft and lightly browned, about 10 minutes. Spoon into a bowl and return the skillet to medium heat. Add the remaining 1 tablespoon butter and cook until melted. Stir in the flour and continue to cook until lightly browned and smooth, 1 to 2 minutes. Pour in the reserved juices from the broiler pan and the remaining 2 tablespoons Worcestershire sauce and cook until smooth and hot.
4. Thinly slice the meat against the grain and lightly season the slices with salt. Divide the sliced meat and sauteed onions among the hoagie rolls. Serve with the pan sauce on the side for dipping.
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