1Preheat the oven to 350 degrees, then heat an ovenproof 12-inch skillet over medium to cook the ground beef with a pinch of salt for 5 to 7 minutes until just pink, using a slotted spoon to transfer the meat to a bowl while leaving one teaspoon of fat in the pan.
2To that same skillet, add the butter, onion, celery, carrot, and a pinch of salt, cooking for 5 minutes before stirring in the garlic and tomato paste for 1 minute, followed by the flour.
3Stir in the broth, red wine, peas, Worcestershire sauce, thyme sprigs, bay leaves, and 1 teaspoon each of salt and pepper, simmering for 7 to 10 minutes until thick like gravy, then remove the herbs, fold the beef back in, and set aside.
4Boil the potatoes in 4 quarts of salted water for 15 minutes until knife-tender, drain, and mash them inside the saucepan with the milk, 4 tablespoons of butter, and 1/2 cup of Parmesan until just creamy.
5Spread the mashed potatoes completely over the beef mixture in the skillet, drizzle with the remaining 1 tablespoon of butter, sprinkle with the remaining 1/4 cup of Parmesan, and bake for 30 minutes until golden and bubbly before letting it stand for 10 minutes to serve.