Cook lentils in a large pot of water until tender but not mushy (about 20-25 minutes). Drain and set aside.
While the lentils cook, place potatoes in another pot, cover with water, and boil until fork-tender (about 15-20 minutes).
Heat olive oil in a large skillet over medium heat. Add onions, carrots, and celery. Cook until vegetables begin to soften (5-7 minutes).
Add mushrooms and garlic to the skillet. Cook until mushrooms release their moisture and begin to brown (8-10 minutes).
Stir in tomato paste, herbs, soy sauce, and vegetable broth. Simmer for 5 minutes.
Fold in the cooked lentils and season with salt and pepper. Remove from heat.
Mash the potatoes with milk and butter until smooth. Season to taste.
Transfer the filling to a baking dish and top with mashed potatoes, creating peaks with a fork.
Bake at 375°F (190°C) for 25-30 minutes until golden brown on top.