Heat the olive oil in a large pot over medium heat. Add the diced onion and cook until it softens, about 5 minutes.
Stir in the minced garlic and sliced mushrooms. Cook until the mushrooms release their liquid and start to brown, about 6 minutes.
Sprinkle the smoked paprika over the vegetables and stir for 30 seconds to toast the spice.
Add the rinsed lentils, vegetable broth, and diced tomatoes with their juice. Bring the mixture to a boil, then reduce the heat to low.
Cover and simmer for 25 minutes, stirring occasionally, until the lentils are tender.
Stir in the chopped kale and cook for another 5 minutes until the greens wilt. Season with salt and pepper before serving.