Heat 2–3 tablespoons water in a large skillet. Add mushrooms, garlic, and onion and water-sauté 5 minutes at medium heat or until onions are tender and mushrooms have cooked down.
Add balsamic vinegar and simmer for an additional minute. Add kale and continue cooking until kale has softened.
Add tomatoes, chickpeas, vegetable broth, thyme, black pepper, and cayenne pepper.
Simmer for 10 minutes, breaking up some of the chickpeas and tomatoes with the back of a spoon.
To prepare polenta, bring water to a boil over high heat, then slowly whisk in cornmeal. When all the cornmeal is added, reduce heat to a very low simmer, cover, and continue cooking until the polenta is smooth and thick, about 10–20 minutes, stirring every 5 minutes. Serve polenta topped with kale and chickpea mixture.