Combine oil, onion, and garlic in a Dutch oven over medium heat. Cook and stir until onion softens, 3 to 5 minutes. Add mushrooms, Japanese eggplant, and buttercup squash sequentially, stirring well after each addition.
Stir tomatoes into the Dutch oven. Stir in red lentils; add pasta shells. Pour in stock; reduce heat to medium-low and simmer, covered, until pasta is almost fully cooked, about 30 minutes. Stir occasionally to ensure pasta doesn't stick to the bottom of the Dutch oven.
Mix rice vinegar, soy sauce, and tomato paste together in a small bowl. Pour into the Dutch oven; stir to combine. Stir in nutritional yeast. Add basil, paprika, and black pepper; stir once more.