Preheat oven to 375°F.
In a large ovenproof sauté or cast iron pan over medium-high heat, heat the oil, and then add the onion, carrot, celery and meat. Cook until the meat is browned, about 8 to 10 minutes.If using lamb you will need to drain the fat then add the broth, tomato paste, and herbs. Simmer until the juices thicken, about 10 minutes, and then add the peas.
You can either pour the mixture into a 1 1/2-quart baking dish, or, what I like to do, is continue to use the same sauté or cast iron pan, thus avoiding having to wash another dish. Set the mixture aside.
While the onion, carrot, celery and meat mixture is sautéing; bring the potatoes to a boil in salted water. Cook until fork tender, about 20 minutes; drain.
Mash the potatoes with the butter, milk, and salt.
Spread the potatoes over the meat mixture, and then crosshatch the top with a fork. This makes a really pretty design and fancies up the dish a bit.
Bake until potatoes are golden brown, 30 to 35 minutes.