Place the peeled and cubed potatoes in a large pot and cover with cold salted water. Boil until fork-tender, about 15-18 minutes. Drain thoroughly and return to the pot to steam dry for a minute or two.
In a large Dutch oven or deep skillet over medium heat, sauté the onion, carrots, and celery in olive oil for 6-8 minutes until softened. Stir in the garlic and cook for 30 seconds until fragrant.
Add the chopped mushrooms and cook for 5-6 minutes until their liquid releases and mostly evaporates. Stir in the cooked lentils, peas, and corn and cook for 2 more minutes.
Push the mixture to one side and cook the tomato paste alone in the empty space for about 1 minute to deepen its flavor. Stir everything together, then add the tamari, vegan Worcestershire sauce, vegetable broth, smoked paprika, thyme, and rosemary. Bring to a simmer and cook uncovered for 5 minutes.
Whisk the cornstarch with cold water to make a slurry. Pour it into the simmering filling while stirring constantly and cook for 2-3 minutes until thickened. Season with salt and pepper to taste, then remove from heat.
Preheat oven to 400°F. Add vegan butter to the drained potatoes and mash until mostly smooth. Gradually pour in the warmed oat milk while mashing until creamy. Stir in nutritional yeast (if using) and season with salt and pepper.
Spread the filling evenly in a 9x13 baking dish. Top with large dollops of mashed potatoes and spread into an even layer, sealing the filling underneath. For a decorative finish, pipe the potatoes instead.
Bake for 20-25 minutes until the filling is bubbling and the top is lightly golden. For extra color, broil for the final 2-3 minutes. Let rest for 10 minutes before serving. Garnish with chives if desired.