Heat oil in a large skillet over medium. Cook onion, chile, and garlic, stirring occasionally, until soft, 8–10 minutes. Add mushrooms, tofu, chili powder, and paprika. Cook, stirring occasionally, until mushrooms soften and are nearly dry, about 5 minutes. Add tomato purée, walnuts, apple cider vinegar, ketchup, and soy sauce. Bring to a simmer and cook until liquid is thick, 12–15 minutes; season with salt. Serve on buns.