Place the cubed potatoes in a large pot and cover with cold water. Bring to a boil over high heat, reduce to a simmer, and cook until fork-tender, about 15-20 minutes. Drain well.

Return the drained potatoes to the pot. Add vegan butter, non-dairy milk, salt, and pepper. Mash until smooth and creamy. Set aside.

Heat olive oil in a large skillet over medium heat. Add the diced onion, minced garlic, carrots, and celery. Cook for 5-7 minutes until the vegetables soften. Sprinkle in flour and stir; cook for 1 minute.

Stir in the tomato paste, lentils, vegetable broth, and dried thyme. Bring the mixture to a simmer. Add salt and pepper to taste.

Mix in the frozen peas and corn; cook for an additional 3-4 minutes until they are heated through and the sauce thickens.

Preheat your oven to 400°F. Transfer the lentil and vegetable filling into a greased baking dish, and spread it evenly. Top with the mashed potatoes, smoothing the surface with a spoon.

Use a fork to create a light pattern on the potatoes for extra browning. Bake for 25-30 minutes until the top is lightly browned. Let cool slightly before serving. Garnish with parsley.