Add oil to a large nonstick skillet and heat over medium-high heat. Add the onions and cook until just beginning to soften. Add the ground lamb or beef and break up with a spatula as it begins to cook. Then stir in the rosemary, thyme, salt, and pepper. Cook for 6 minutes or until the lamb is browned. Remove all but 2 tablespoons of fat from the skillet.

Stir in the Worcestershire sauce and garlic and let cook until fragrant, about 30 seconds. Then quickly stir in the flour and tomato paste, stirring until smooth with no clumps.

Add the broth and frozen peas and carrots; bring to a boil before reducing to a simmer. Continue to simmer for 5 to 8 minutes, or until the filling has thickened. Set aside.

Preheat the oven to 400 degrees F.

While the filling is cooking, add potato chunks to a pot of cold water that's covering the potatoes by about an inch. Partially cover and bring to a boil over high heat. Cook until the potatoes are fork-tender.

Drain the potatoes and return to the pot. Let them sit for one minute to absorb any remaining water. Add the butter, 1/2 cup of milk or half and half, and salt and pepper. Mash the potatoes until smooth and creamy, adding milk as needed. Stir in the spring onions and Parmesan cheese.

Transfer the meat mixture into a 9 x 9-inch pan or deep 10-inch pie plate (as I've done). Spread into an even layer. Spoon the potatoes over the filling and swirl to make a nice pattern.

Bake uncovered for 20 to 25 minutes, or until the mash is beginning to brown and the filling is beginning to bubble around the edges. Remove from the oven and let rest for a few minutes before serving.