Bring a medium saucepan of water to a boil. Add the lentils to the water along with a fat pinch of salt. Simmer the lentils until they’re just-tender, about 20 minutes. Drain and set aside.
In a braiser or medium soup pot, heat the olive oil over medium heat. Add the shallots to the pot and stir. Cook the shallots until slightly softened and translucent, about 4 minutes. Add the sliced mushrooms to the pot and let them sit for a full minute. Stir the mushrooms up and season with pepper. Let the mushrooms sit in the pot for another full minute.
Stir the mushrooms until they start glistening slightly. Season the mushrooms liberally with salt. Add the garlic and thyme to the pot and stir. Once the garlic is fragrant, about 30 seconds, add the white wine and tamari to the pot. Stir the mushrooms. Add the drained lentils, vegetable stock and plant milk to the pot. Stir and bring the mixture to a boil.
Ladle half of the mushroom mixture into your blender pitcher, ensuring that you include enough of the liquid. Blend on high until completely smooth. Scrape the creamy blended mushroom mixture back into the pot. Add the sliced kale and stir. Bring the mixture to a boil and check it for seasoning. The texture should be like a thick and creamy stew and the kale should be slightly wilted..
Serve the creamy French lentils hot with toasts or other accompaniments of your choosing.