Preheat the oven to 400°F (200°C). Lightly grease a 9x13-inch baking dish and set it aside.
Place the potatoes in a large pot and cover with cold salted water. Bring to a boil and cook until fork tender, about 15 to 18 minutes. Drain well, then mash with the butter, warm milk, sour cream, garlic powder, salt, and black pepper until smooth and creamy. Set aside.
Meanwhile, heat the olive oil in a large skillet over medium heat. Add the onion, carrots, and celery, cooking until softened, about 7 to 8 minutes. Stir in the mushrooms and continue cooking until they release their moisture and begin to brown.
Mix in the garlic, tomato paste, thyme, rosemary, smoked paprika, Worcestershire sauce, and soy sauce. Cook for another minute until fragrant.
Sprinkle the flour over the vegetables and cook, stirring constantly, for about 1 minute before adding the broth. Slowly pour in the vegetable broth while stirring constantly to create a smooth gravy.
Fold in the cooked lentils and simmer for about 8 to 10 minutes until the filling thickens. Stir in the frozen peas and chopped parsley, then season with salt and pepper to taste.
Spread the lentil mixture evenly into the prepared baking dish. Spoon the mashed potatoes over the filling and gently spread them into an even layer. Use the back of a spoon or a fork to create ridges across the surface for extra browning.
Bake for 25 to 30 minutes, or until the filling is bubbling around the edges and the potato topping is lightly golden. For deeper color, broil for the final 2 to 3 minutes while watching carefully.
Let the shepherd's pie rest for about 10 minutes before serving so the filling can set.