Soak lentils in 6 cups of water and 1 tablespoon of salt for 3 to 4 hours and up to 24 hours. Drain and turn into a large skillet. Add 1 cup of water, cover skillet, and turn burner on high. Bring to a boil; cook until water has evaporated and lentils are just tender, about 3 minutes. Turn into a large bowl. Remove 3 cups of the lentils and reserve for another use.
Meanwhile, bring whole potatoes to boil over medium-high heat in a small Dutch oven; cover and reduce heat to medium-low. Cook until tender, 25 to 30 minutes, depending on potato size. Drain potatoes and mash right in the pot, adding pant-based milk, a generous sprinkling of salt and pepper, and the plant-based butter; set aside.
Heat 2 tablespoons of oil over medium-high heat in a large (12-inch) skillet. When oil is hot, add mushrooms; sauté until tender, 5 to 7 minutes, seasoning with salt and pepper towards the end of cooking. Transfer mushrooms to bowl with lentils.
Heat oven to 400 degrees. Return skillet to the burner and heat remaining 2 tablespoons of oil. Add onions and carrots; sauté until just tender, about 5 minutes. Stir in thyme, and then flour; cook until flour is evenly incorporated. Add wine, followed by broth, stirring until mixture is smooth. Stir in tomato paste and Worcestershire; continue to cook until mixture has thickened to thin sauce consistency. Return lentils and mushrooms and cook to heat through. Adjust seasonings, including salt and pepper. Turn mixture into a 13- by 9-inch or similar size casserole dish. Cover with mashed potatoes; bake until filling is bubbly, 25 to 30 minutes. Turn oven to broil; cook until potato topping is golden brown, 2 to 3 minutes longer. Serve with extra Worcestershire sauce for drizzling.