Meanwhile, prepare the filling: In a 10-inch oven-safe skillet with high sides or an enameled cast-iron braiser over medium-high heat, bring the lentils, thyme and 2 cups of the broth to a simmer with 1 teaspoon salt. Reduce the heat and continue to cook the lentils, partly covered, until they are tender and most of the liquid is absorbed, about 20 minutes. Remove thyme sprigs and transfer the lentils to a bowl.

Finish the topping: In a small saucepan or a microwave oven, heat 6 tablespoons of the butter and milk together until butter melts. Drain potatoes, and rice or mash them until smooth in a bowl.

Mix in the hot butter mixture and sour cream just until blended. Stir in ½ cup of the Parmigiano. Season to taste with salt, pepper and nutmeg. Cover and set aside.

Finish the filling: Melt the butter in the 10-inch skillet over medium-high heat. Sauté onion for 6 or 7 minutes, season to taste with salt and pepper, add the mushrooms and cook, stirring often with a wooden or silicone spoon, until they are deep golden brown, about 8 to 10 minutes. Reduce the heat to medium and add leeks, carrots and garlic, and continue to cook partially covered until tender, another 10 minutes.

Add the tomato paste and stir, cooking until it is well combined, another 2 to 3 minutes. Sprinkle flour over the mixture, stir and cook for 1 minute. Add remaining 1½ cups broth, cooked lentils and peas, and cook until thickened. Stir in lemon juice to taste.

Heat the oven to 375 degrees.
Spread mixture to a casserole dish, and top the mixture with dollops of the mashed potatoes, then spread them out over the top. Sprinkle with breadcrumbs, if desired, and the remaining 1/2 cup of Parmigiano.

Transfer to the oven and, if the mixture is at the top edges of your pan, set a foil-lined baking sheet underneath the pan to catch any drips. Bake the pie until the potatoes have begun to brown and the edges are bubbling, about 30 minutes. Let stand at room temperature for 15 minutes before serving.