Pierce the steak all over with a fork and place in a resealable plastic bag. Whisk together the tomato paste, balsamic vinegar, 1/2 cup of the beef broth and 2 tablespoons of the Worcestershire 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 in the refrigerator for 4 to 6 hours.
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 bag and discard the marinade. 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 prepared pan and place the meat on top of the broiler pan or rack. Broil, turning once, until a thermometer inserted in the center registers about 125 degrees F, about 6 minutes per side for medium-rare. Let the meat rest on the rack for 5 minutes 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.
Meanwhile, heat 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 set aside.
Pour the reserved juices from the broiler pan and the remaining 2 tablespoons Worcestershire sauce into a small pot and cook over medium heat until hot.
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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