Make the mole: Heat 2 tablespoons of the oil or lard in a medium-large (4 to 6 quart) heavy pot or Dutch oven over medium. Add the anchos, onion, and garlic and fry for 6 or 7 minutes, stirring frequently, until the onion and garlic have softened and browned and the chile is toasty-looking and aromatic. Scrape it into a blender jar and set the pan aside.
If using fresh tomatoes, roast them on a baking sheet below a very hot broiler until splotchy-black and soft, about 5 minutes, then flip them and roast the other side. Cool, then peel and transfer to the blender with the ancho mixture. If using canned tomatoes, simply scoop them into the blender. Add the peanuts, white bread, chipotles, allspice, cinnamon, and 1 ½ cups of thechicken broth. Blend until smooth, stirring and scraping down the sides of the blender jar and adding a little more liquid if needed to keep everything moving through the blades. You will know it’s ready when a drop of the puree rubbed between your fingers doesn’t feel gritty. Press the mixture through a medium mesh strainer into a bowl.
Heat the remaining 2 tablespoons of oil or lard in the pot over medium-high. When hot enough to make a drop of the sauce sizzle fiercely, add the strained mixture all at once. Stir constantly for 7 or 8 minutes until it has reduced by two thirds its original volume and darkened to a thick, rusty-orange paste (about the consistency of tomato paste)—this is the critical step that fuses these disparate flavors into a harmonious sauce.Finally stir in the remaining 2 cups of chicken brothand the Mexican chocolate. Partially cover the pot, turn the heat to low and let it simmer, stirring occasionally, for as long as you have—½ hour is good, but an hour is better; making the mole a day ahead and refrigerating produces the best flavor. If necessary thin the sauce with a little more broth to keep it the consistency of a cream soup. Taste and season with salt (about 2 ½ teaspoons if using unsalted broth) and about ½ teaspoon sugar (to bring out the fruitiness of the chiles).If you’ve made the mole previously and refrigerated it, rewarm it and adjust the consistency and seasonings if necessary. Keep warm while you prepare the quail.
Grill the quail: Heat a cast iron or nonstick grill pan to between medium and medium-high. Pat the quail dry with a paper towel and cut off the legs with a knife. Tuck the wings under the breasts so that they almost lay flat. Drizzle with olive oil and sprinkle with salt.
Lay the quail breasts and legs on the hot grill pan skin side down and cook about 6 to 7 minutes, checking once or twice to ensure that they are not browning too quickly. Flip the quail and continue cooking until the breasts are just barely done and the leg meat will separate from the bone quite easily when you squeeze a leg between two fingers, 4 to 6 minutes more.
Remove to a plate and keep warm in a low oven while you set up your plates. Ladle a generous 1/3 cup of the hot earthy-colored mole onto each of 6 warm dinner plates. Set quail over the sauce. Garnish with chopped peanuts and sprigs of parsley.