Prepare the Lentil Filling:

1. Sauté Aromatics: Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium heat. Add the chopped onion, diced carrots, and diced celery. Cook, stirring occasionally, for 8-10 minutes, or until the vegetables have softened.
2. Add Garlic and Mushrooms: Stir in the minced garlic and sliced/chopped mushrooms. Cook for another 5-7 minutes, until the mushrooms have released their liquid and started to brown.
3. Incorporate Lentils and Flavor Base: Add the rinsed green lentils, vegetable broth, diced tomatoes (undrained), tomato paste, soy sauce (or tamari), dried thyme, dried rosemary, and smoked paprika (if using). Stir well to combine.
4. Simmer the Filling: Bring the mixture to a boil, then reduce the heat to low, cover, and simmer for 35-45 minutes, or until the lentils are tender but not mushy. Stir occasionally to prevent sticking.
5. Thicken the Filling: If using flour, sprinkle it over the lentil mixture and stir vigorously for 1-2 minutes until it’s fully incorporated and the sauce begins to thicken. If using a cornstarch slurry, whisk it in and simmer for another 1-2 minutes until thickened.
6. Add Frozen Vegetables: Stir in the frozen peas and frozen corn (if using). Cook for another 5 minutes, or until heated through.
7. Season: Taste the filling and adjust seasoning with salt and freshly ground black pepper as needed. Set aside.

Prepare the Mashed Potato Topping:

1. Boil Potatoes: While the lentil filling is simmering, place the peeled and quartered potatoes in a large pot and cover them with cold, salted water. Bring to a boil over high heat, then reduce heat to medium-low and simmer for 15-20 minutes, or until the potatoes are very tender when pierced with a fork.
2. Drain and Mash: Drain the potatoes thoroughly in a colander. Return them to the hot pot (off the heat) for a minute to allow any excess moisture to evaporate.
3. Add Dairy/Fats: Add the milk, butter, salt, white pepper (if using), and nutmeg (if using) to the potatoes.
4. Mash to Perfection: Mash the potatoes using a potato masher until smooth and creamy. Be careful not to over-mash, as this can make them gluey. For an extra smooth mash, you can use a potato ricer.
5. Enrich (Optional): If using, stir in the egg yolk for a richer topping that browns well.

Assemble and Bake the Shepherd’s Pie:

1. Preheat Oven: Preheat your oven to 375°F (190°C).
2. Assemble: Pour the prepared lentil filling into a 9×13 inch baking dish or a similar-sized casserole dish. Spread it out evenly.
3. Top with Mash: Carefully spoon the mashed potato topping over the lentil filling, spreading it gently to cover the entire surface. You can use the back of a spoon to create swirls or a fork to create ridges, which will crisp up nicely.
4. Bake: Place the baking dish in the preheated oven and bake for 25-30 minutes, or until the filling is bubbly around the edges and the potato topping is lightly golden brown. If you want a browner topping, you can place it under the broiler for the last 2-3 minutes, watching carefully to prevent burning.
5. Rest: Remove the Shepherd’s Pie from the oven and let it rest for at least 10-15 minutes before serving. This allows the filling to set slightly and makes it easier to serve.