Preheat the oven to 375°F and spray a 2-qt baking dish with non-stick spray.
Add the olive oil to the skillet over medium-high heat.
Once hot, add the onion, garlic, and carrot. Cook until soft, about 3 minutes.
Add the mushrooms and tomato paste and cook for 2 more minutes.
Reduce to medium-low heat and stir in the flour, salt, black pepper, paprika, and thyme. Cook for 2 minutes, stirring continuously.
Pour in the broth and soy sauce. Add the lentils and peas and cook for 5 minutes until the sauce thickens.
Pour the lentil filling into the prepared casserole dish and spread it into an even layer. Set aside.
Add the potatoes to a large stock pot. Cover the potatoes with cold water and bring to a boil. Cook until fork-tender, about 15 minutes.
Drain the potatoes and return them to the stockpot. Place the pot onto the stove with the burner turned off. Allow the potatoes to dry out for 5 to 7 minutes.
Microwave almond milk for 30 seconds. Add half of the plant milk, vegan butter, garlic powder, and salt to the pot with the potatoes.
Mash with a potato masher. Add more almond milk if needed while mashing.
Spread the mashed potatoes over the lentil filling evenly.
Bake the vegan lentil's shepherd's pie for 20 to 25 minutes or until the mashed potato topping is lightly golden.