Heat a large pan over low-medium heat. Add the oil and carrots, and cook until the carrots begin to soften, about 4-5 minutes. Add the onions, and continue to cook until the onions become translucent, another 5-7 minutes. Next add the garlic, and cook for an additional 30 seconds.

Place the beef or lamb in the pan along with thyme, rosemary, salt, and pepper. Break apart the beef as it cooks. Continue cooking until the beef is browned and cooked through.

Sprinkle flour over the beef, and stir to combine. Cook for 5 to 6 minutes then add the beef broth, red wine, tomato paste Worcestershire sauce, and bay leaves. 

Bring the mixture to a rapid simmer, then lower the heat to a slow simmer. Cook for 25 to 30 minutes until the liquid reduces to a gravy. Add the frozen peas and additional salt and pepper if needed.

Transfer the mixture to the prepared baking dish. Allow the filling to cool for 30 minutes to an hour in the refrigerator before topping with mashed potatoes.

Preheat the oven to 350°F.

Cook the potatoes in a large pot of boiling salted water until tender. Drain and transfer to a large bowl. 

Add the butter, and use a potato masher to mash until the butter melts. Pour in the milk and heavy cream. Continue to mash until smooth and creamy, then fold in the cheese. The potatoes should be spreadable and not too thin. Use additional milk if necessary to reach the desired consistency.

Spread the potatoes over the top of the meat filling. Use a fork to scrape texture into the potatoes. Drizzle with melted butter.