Gently scrape off the dark gills on the underside of the portobello caps with a spoon, then roughly chop the caps. Combine the portobello mushrooms, cremini mushrooms, Worcestershire and balsamic vinegar in the bowl of a food processor and pulse until the mushrooms are very finely chopped; work in batches if necessary. Set aside.
Heat the oil in a large Dutch oven or large pot over medium heat. Once the oil shimmers, add the onions, bell peppers, jalapeno and garlic and cook, stirring frequently, until the vegetables begin to soften, about 8 minutes. Add the chili powder, cumin, salt and black pepper and cook, stirring, until the spices are fragrant. Add the tomato paste and cook, stirring, until the vegetables are coated well, about 1 minute. Stir in the finely chopped mushrooms and cook, stirring, until all of the liquid they release has evaporated, about 5 minutes.
Stir in the shiitake mushrooms, tomatoes, black beans, cannellini beans, kidney beans, Parmesan rind, and 4 1/2 cups of water. Bring to a simmer and cook until the flavors meld and the mushrooms are tender, about 30 minutes. The chili can be made up to 2 days ahead and reheated.
Serve the chili with scallions, sour cream, grated cheese and cornbread on the side.