Warm the olive oil in a 12-inch ovenproof sauté pan over medium-high heat until it shimmers. Use a baking dish later if the pan cannot go into the oven.

Add the onions with a couple pinches of salt and cook for 10 to 12 minutes, stirring now and then. Most of the onions should be browned; add a splash of water and lower the heat slightly if they color too quickly.

Stir in the garlic, thyme, and rosemary for 2 minutes. The herbs should smell fragrant and the garlic should soften without browning.

Add the tomato paste and cook for 2 to 3 minutes, stirring frequently. Lower the heat to medium if needed; the paste should deepen to a rusty brick-red color.

Pour in the red wine and scrape the browned bits from the pan as it cooks for 3 to 4 minutes. Continue until most of the liquid has evaporated and the sharp alcohol aroma fades.

Add the vegetable broth, lentils, bay leaves, and paprika, then bring the mixture to a boil. Reduce to a rapid simmer with steady bubbles across most of the pan and cook for about 30 minutes, until the lentils are just tender and most liquid is absorbed.

Lower the heat and stir in the tahini, tamari or soy sauce, balsamic vinegar, and vegan Worcestershire sauce (if using). The filling should look creamy and thick; season generously with salt and black pepper to taste.

Peel the potatoes, rinse them under cold water, and cut each potato into eighths. Even pieces cook at the same pace and mash more smoothly.

Cover the potatoes and cauliflower with water in a large saucepan or Dutch oven, add 1 1/2 teaspoons kosher salt, and place the rosemary sprig on top. Boil for about 15 minutes, until both vegetables are very soft and yield easily to a fork.

Drain the vegetables thoroughly and discard the rosemary sprig. Pat the potatoes dry, then squeeze excess moisture from the cauliflower through a fine-mesh sieve or clean dish towel.

Return the vegetables to the warm pan or move them to a large bowl, then add the softened butter, coconut milk, and nutritional yeast. Mash with a potato masher or immersion blender until smooth, then season with salt and pepper to taste.

Heat the oven to 375°F/190°C and level the lentil filling in the ovenproof skillet or a large 3 quart/3 liter baking dish. Spoon the mashed topping over the filling and spread it to cover the surface completely.

Drizzle the top with extra virgin olive oil and add a few pinches of salt and pepper. Drag fork tines across the mash to form ridges that will brown under the broiler.

Bake for 20 minutes, until the filling bubbles around the topping. Switch on the broiler and cook for a few minutes until the peaks become golden brown.

Finish with more olive oil and chopped flat-leaf parsley, if desired, then let the pie rest for at least 20 minutes. The filling firms as it rests, making portions neater and easier to serve.