Season beef with a pinch of salt and black pepper. Heat vegetable oil in a heavy pot over high heat. Cook and stir beef in hot oil until browned, 5 to 8 minutes.
Stir garlic, vinegar, oregano, 1 1/2 teaspoons salt, thyme, rosemary, 1 teaspoon black pepper, bay leaf, and red pepper flakes into beef. Pour enough chicken broth into beef mixture to cover the meat by 1 inch and bring to a simmer.
Cover pot with a lid, reduce heat to low, and simmer until meat is fork-tender, 1 to 1 1/2 hours.
Transfer meat with a strainer or slotted spoon to a separate pot; pour about 1/4 cup of meat broth into pot. Use a wooden spoon to gently break meat into smaller chunks. Cover pot with a lid or aluminum foil and keep warm.
Skim excess grease from top of broth remaining in the first pot; season with salt and pepper to taste. Cover pot with a lid or aluminum foil and keep broth warm.
Lay halves of a roll out on a work surface and spoon 2 to 3 tablespoons meat broth over each half. Top bottom half of roll with a generous portion of meat and a spoonful of pickled vegetables. Place tops on sandwich. Repeat with remaining buns, broth, meat, and pickled vegetables to make 3 more sandwiches.
Spoon hot meat broth into ramekins and top each ramekin with 1/2 teaspoon parsley. Serve sandwiches with hot broth for dipping.
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