In a large nonstick skillet cook the meat over medium heat, stirring occasionally, until brown, about 5 minutes. Transfer the meat to a plate. Drain any fat remaining in the skillet.
Heat the oil in the skillet over a medium heat. Add the onions and carrots and cook, covered, until the vegetables begin to soften, about 8 minutes. Raise the heat to moderately-high. Add the mushrooms and thyme and cook, uncovered and stirring occasionally, until the mushrooms are soft and their liquid has evaporated, about 8 minutes longer. Return the meat to the pan. Add the flour and cook, stirring, for 2 minutes. Stir in the broth, 1/2 teaspoon of the salt and 1/8 teaspoon of pepper and bring to a simmer being sure to scrape up any brown bits from the bottom of the pan. Stir in the peas. Pour the mixture into a 12-cup shallow baking dish (about 11 by 9 inches).
Preheat the oven to 350 degrees F. Scrub the potatoes and cut into 2-inch pieces. Arrange the potatoes in a steamer basket, and steam for 10 minutes. Add the cauliflower to the basket and cook until the potatoes and cauliflower are tender when pierced with the tip of a knife, about 15 minutes longer. Mash the vegetables with a potato masher until smooth. Heat the milk, butter, the remaining 1/2 teaspoon of salt and a pinch of pepper and stir into the potato mixture.
Spread the potato mixture on top of the meat and bake until heated through, about 25 minutes.