Make the roasted garlic: Toss the 12 cloves garlic with the olive oil and 1 teaspoon water in a microwave-safe bowl. Cover, leaving a vent, and microwave until soft, about 90 seconds. Let cool, covered.
Prepare the chili: Preheat the broiler. Cook the shallots in the olive oil in a large saucepan over medium-low heat until caramelized, about 20 minutes. Meanwhile, place the chiles on a foil-lined broiler pan and broil until charred on all sides, turning, about 8 minutes. Transfer to a bowl, cover with plastic wrap and let cool. Peel the peppers with your fingers or a paring knife. Stem, seed and chop.
Add the minced garlic to the shallots and cook until fragrant, about 2 minutes. Add the flour and cook, stirring, until toasted, about 3 minutes. Increase the heat to high and add the wine; simmer 2 minutes, scraping up any browned bits from the pan. Add the roasted chiles, 6 cups broth, the chicken, chili powder, cayenne, beans, and salt and black pepper to taste. Return to a simmer.
Meanwhile, squeeze the soft pulp from the roasted garlic into a blender or food processor. Add the remaining 1 cup broth and process until smooth. Add to the chili and simmer until thickened, about 30 minutes, adding the spinach during the last 5 minutes. Add the paprika and cream and season with salt and pepper. Garnish with the cheese.