Preheat your oven to 375°F. While the oven’s warming up, make the mashed potatoes: place peeled and cut potatoes in a pot and cover with water. Generously salt the water and bring to a boil over high heat. When boiling, reduce the heat to maintain a simmer and cook for 15-20 minutes or until fork tender.
While the potatoes are cooking, make the pie filling. Heat a bit of oil over medium-high heat in a large skillet. When the oil is hot, add the diced onions and brown slightly, 2 minutes or so. Add the garlic, carrots, and celery. Cook until slightly tender, 2-3 minutes. Add the ground beef and cook, breaking up into pieces. When browned and cooked through, stir in the flour.
Add the broth, tomato paste, soy sauce, thyme, and rosemary. Stir and cook over medium low. Stir in the the corn and peas. Mix well, simmering, until it thickens slightly. Taste and season with salt and pepper, then scoop into the bottom of an oven-safe casserole dish.
When the potatoes are cooked, drain well and return to the pot. Add the roasted garlic. Over very low heat, mash the potatoes. Stir in 2 tablespoons butter and the sour cream. Stir in the milk until desired consistency. Taste and season generously.
Top the meat portion with the mashed potatoes and dot with the remaining 1 tablespoon of butter.
Bake the casserole (place on a baking sheet to catch any drips) until the potatoes are golden brown and crispy, about 30 minutes. If needed, broil for 5-8 minutes for an extra golden potato top. Enjoy warm.