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, then crosshatch the top with a fork. Bake until golden, 30 to 35 minutes.
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