Place the peeled potato quarters in a medium saucepan, cover with cold water by an inch, add 1 teaspoon salt, bring to a boil, then simmer until fork-tender, about 20 minutes.

While the potatoes cook, set the oven to 400 °F (200 °C).

In a large sauté pan over medium heat, melt 4 tablespoons butter, add the chopped onion, and cook until softened, 6–10 minutes; incorporate diced carrots at this stage if using.

Stir in peas or corn near the end of the onion’s cooking time, then add the ground beef, breaking it up as it browns; drain off excess fat, leaving roughly 1 tablespoon.

Season the meat with salt, pepper, Worcestershire sauce, and pour in the beef broth; bring to a gentle simmer and let it cook uncovered for 10 minutes, adding a splash more broth if the mixture begins to dry.

Drain the finished potatoes, combine with the remaining 4 tablespoons butter, mash until smooth, and adjust seasoning with salt and pepper.

Spread the beef-vegetable filling evenly in a 9 × 13-inch baking dish; spoon the mashed potatoes over the top, roughing up the surface with a fork to create peaks for browning.

Bake for about 30 minutes, until the topping is browned and the filling bubbles at the edges; if deeper color is desired, briefly broil, monitoring closely and avoiding glass bakeware under direct broiler heat.