Preheat a large, heavy skillet, preferably cast-iron, over medium-high heat. Season the meat with salt and pepper.
Drizzle a little oil into the hot pan. Add the beef and cook, turning once, until nicely browned, about 6 minutes.
While the burgers cook, in a small bowl, mash together the butter, shallot, garlic, tomato paste, parsley, Worcestershire and oregano with a fork.
Remove burgers from the skillet and tent loosely with folk to keep warm. To the skillet, stir in the vermouth over medium-high heat and scrape up any browned bits. Add the butter mixture and let simmer for 2-3 minutes.
Layer the mozzarella on the bottoms of the burger buns. Top with a burger and one-quarter of the butter-tomato mixture so it melts down over the meat and is soaked into the bun bottoms. Cover with the bun tops and serve.