Place dried mushrooms and 2 cups chicken stock in a small pot over medium-low heat.
Heat the extra-virgin olive oil in a Dutch oven or high sided skillet over medium to medium-high heat. Add garlic, crushed pepper, onions, carrots, celery, peppers, and eggplant to the pot as you chop them. Add the bay leaf, season with salt and pepper and cover the pot to sweat the vegetables out a little faster. Stir occasionally for 10 to 12 minutes then uncover and add the balsamic vinegar.
Remove the reconstituted dried mushrooms to the pot with a slotted spoon. Pour in all but the last few tablespoons of mushroom stock. The last of the stock can hold a little loose grit from the dried mushrooms. Add tomatoes and break up with a wooden spoon. Simmer over low heat the last few minutes before serving. Adjust salt and pepper.
Heat the remaining 2 cups stock with 1 cup milk, golden raisins and rosemary over medium high heat. Whisk in polenta and stir 3 minutes until thickened. Remove from heat and stir in honey, butter and cheese.
Pile some polenta in shallow bowls and spoon the stew around the polenta.;