Peel the potatoes and cut them into 3 cm (1 1/4 inch) cubes.
Put the potatoes in a pot of cold salted water, bring to a boil over high heat and simmer over medium heat until very tender, about 15 minutes. Drain and mash with the soy milk, half the oil and half the salt.
While the potatoes cook, heat the rest of the oil in a large deep skillet over medium heat.
Add the onion, carrot, celery and mushrooms and cook, stirring, until soft and the mushroom liquid has evaporated, about 8 minutes. Add the garlic and tomato paste and stir for 1 minute.
Add the lentils, broth, thyme and soy sauce. Bring to a boil, then reduce to medium-low and simmer, stirring now and then, until the lentils are tender and the mixture is thick, about 25 minutes. Stir in the peas and the rest of the salt and pepper.
Preheat the oven to 200°C (400°F) with a rack in the centre while the filling simmers.
Spread the filling in a 23 x 33 cm (9 x 13 inch) baking dish and spoon the mash over it. Spread to the edges and rough up the top with a fork so it crisps.
Bake for 20 minutes, until bubbling at the edges and golden on top. Put a tray underneath to catch drips.
Let it stand for 10 minutes before cutting so it holds together.
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