Char peppers over flame burner or under a broiler until blackened all over. Place charred peppers in a bowl and cover with plastic wrap, cool to handle, seed and peel peppers and thinly slice.
Season the chicken liberally with salt and pepper, heat a tablespoon extra-virgin olive oil, a turn of the pan, in a Dutch oven over medium-high heat. Add the chicken and brown a few minutes on each side, then remove to a plate. Add the remaining extra-virgin olive oil and mushrooms, until browned. Add the onions and garlic, season with salt and pepper, stir occasionally for 8 to 10 minutes, then deglaze the pan with sherry or wine. Stir in the prepared tomatoes, peppers, and paprika. Cook for 10 minutes. Add the chicken and simmer 10 minutes more. Turn off the heat and cool the chicken, then cover and store for a make-ahead meal.
Cook's Note: Reheat over medium heat, covered, then simmer uncovered until ready to serve.
When the chicken is hot, bring the saffron and prepared stock to a boil. Whisk in the polenta and cook until it begins to pull away from the pot. Season with salt and stir in grated cheese.
Serve the polenta in shallow bowls and top with Spanish Cacciatore, and garnish with parsley.
Parmigiano and Herb-Fortified Stock
For the stock:
Place the cheese rind, herb bundle, onion, celery, carrots, lemon rind, bay leaves, chicken stock, and water into a pot and bring to a bubble, then reduce the heat to a simmer. Simmer at least 1 hour, and then remove the rind, herb bundle, and vegetables with a slotted spoon or strainer.
For the tomatoes:
Heat the oven to 500 degrees F.
Arrange the tomatoes on a baking sheet or baking sheets in a single layer. Scatter the garlic among the tomatoes, dress with extra-virgin olive oil to coat and season with salt and pepper. Roast the tomatoes until they burst and skins split and begin to char, about 30 minutes. Cool the tomatoes until cool enough to handle and peel. Place the tomatoes in a bowl.