Begin cooking your preferred pasta according to instructions on packaging. The sauce will take ~7 minutes to prepare, so plan accordingly.
In a large pan, add extra virgin olive oil and when it is hot, add mushrooms. Cook for ~2-3 mins, then add garlic and onions, together with salt & pepper and thyme leaves (not the sprigs themselves!).
After about 1-2 mins, deglaze the pan with soy sauce and balsamic vinegar and reduce heat to a simmer.
Add baba ghanoush and nutritional yeast and once dissolved, add mushroom broth or pasta water (i.e., the water your pasta is cooking in). This should start to thicken to about half its original volume pretty quickly, so keep an eye on it.
Once pasta is cooked, add to pan with sauce, together with kale, and incorporate. Add additional pasta water if pasta gets too dry. Add extra black pepper and a drizzle of olive oil before serving.