Rinse the rice, and bring it to a boil with 1 ¼ cups of water. Cover, reduce heat to low, and simmer for 35 to 40 minutes (until water is absorbed).
Peel the potatoes and cut them in half. Put in a pot, cover with water, and bring to a boil. Reduce heat to medium, and cook for about 30 minutes, until potatoes are easily pierced with a knife.Don't drain the potatoes as we'll use some of the water to make the mash potato.
Peel and dice the onion, garlic, celery, and carrot ; rinse the lentils.
In a pot, heat up the olive oil on medium heat, and saute the onion for a few minutes. De-glaze with the wine, and add the garlic and the celery. Cook for a couple more minutes.Add the carrot, thyme, lentils, and cover with water. Bring to a boil, and simmer with a lid on for about 30 minutes, until lentils are cooked. Drain and remove the thyme stems.
Preheat the oven to 400°F - 200°C.
When the rice and the lentils are cooked, mix them together in a bowl. Blend half of it with a stand or hand blender. Put everything back together into the bowl, add the tomato paste, ½ teaspoon of salt, and a pinch of pepper.
Mash the potatoes with a fork, or in the bowl of your stand mixer with the paddle attachment, and add some cooking water to loosen it up (or plant-based milk if you prefer).
Grease your baking dish, spread the rice and lentil mixture evenly on the bottom, and the mashed potatoes on top. Draw lines with a fork.
Bake for 20 minutes at 400°F - 200°C.Sprinkle with vegan parmesan, and broil for a couple more minutes.