Preheat oven to 375°F. In a large sauté pan over medium-high heat, heat oil, then add the onion, carrot, and meat. Cook until browned, 8 to 10 minutes. Drain the fat and add the broth, tomato paste, and herbs. Simmer until the juices thicken, about 10 minutes, then add the peas. Pour the mixture into a 1 1/2-quart baking dish; set aside.
Meanwhile, bring the potatoes to a boil in salted water. Cook until tender, about 20 minutes; drain.
Mash the potatoes with the butter, half and half, and salt. Spread them over the meat mixture (after making sure the meat reaches the correct temperature), then crosshatch the top with a fork. Bake until golden, 30 to 35 minutes.