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1
Preheat the oven to 375 degrees; use a 9x13 casserole dish.
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2
Place the potatoes in a large steamer basket over boiling water in a pot; cover and steam for 10 minutes on medium-high heat. Add the cauliflower to the basket; cover and steam for about 15 minutes, until the potatoes and cauliflower are tender when pierced with the tip of a knife.
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3
Remove the steamer basket and drain the pot; return the pot to the stove top (off the heat). Transfer the hot vegetables to the pot; use a potato masher to mash them to the consistency you prefer.
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4
Combine the milk and butter in a microwave-safe measuring cup; microwave on LOW just until heated through and the butter has melted (do not boil). Stir into the mashed mixture, along with 3/4 teaspoon of the salt.
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5
Meanwhile, cook the ground lamb in a large nonstick skillet over medium heat, stirring occasionally, until browned, about 5 minutes. Transfer the meat to a plate. Drain/wipe out any fat remaining in the skillet.
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6
Heat the oil in the (same) skillet over medium heat. Add the onion, stirring to coat; cover and cook for about 8 minutes, until softened. Add the mushrooms and thyme; cook uncovered, stirring occasionally, until the mushrooms are soft and their released moisture has evaporated, about 8 minutes.
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7
Return the ground lamb to the skillet. Sprinkle the flour over the skillet mixture; cook over medium heat, stirring, for 2 minutes, then pour in the broth and add the remaining 1/2 teaspoon of salt and the pepper, stirring to incorporate. Once the mixture is barely bubbling, use a spatula to dislodge any browned bits from the bottom of the pan, then stir in the peas. Remove from the heat.
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8
Distribute the meat mixture evenly in the bottom of the baking dish or casserole, then spread the potato-cauliflower mash evenly over the meat-mushroom filling, making sure to cover it completely. Place the baking dish or casserole on a baking sheet (to catch any drips from bubbling filling); bake for 20-25 minutes (top light brown).
