Tip: To stale the bread for making the sauce, simply leave the slices out for a few hours; they should feel dryish on the surface but not brittle
In a large pot, combine the chicken, onion, garlic, parsley sprigs, 1½ teaspoons salt, ½ teaspoon black pepper and 7 cups water. Bring to a simmer over medium-high, then cover, reduce to low and cook, occasionally turning the chicken, until the thickest part of the largest breast (if using) reaches 160°F, 20 to 25 minutes, or the thickest part of the largest thigh (if using) reaches 175°F, 25 to 30 minutes. Using tongs, transfer the chicken to a large plate; set aside until cool enough to handle.
While the chicken cools, strain the broth through a fine-mesh sieve set over a medium bowl. Transfer the garlic cloves to a blender jar; discard the remaining solids. Using tongs or a slotted spoon, lower each slice of bread into the broth just until fully moistened; using your hands, gently squeeze to remove excess liquid and add the bread to the blender.
Measure 3 tablespoons walnuts and set aside for garnish; add the remainder to the blender. Start the blender, then remove the center cap on the blender lid. With the blender running on high, stream in about ¼ cup broth and puree until the mixture is smooth and has the consistency of pourable yogurt; leave the sauce in the blender and set aside.
Shred the chicken into bite-size pieces, discarding the skin and bones. Add the meat to a large bowl. Measure 1 cup of the sauce and add to the chicken along with the lemon juice; toss. If the mixture is too thick, stir in additional broth a few teaspoons at a time to thin. Taste and season with salt and black pepper.
Onto a serving platter, spoon the remaining sauce and spread in an even layer. Spoon the chicken mixture on top and sprinkle with the chopped parsley and reserved walnuts.
In an 8-inch skillet over medium, heat the oil and butter until the butter melts. Add the Aleppo pepper and cook, swirling the pan, until the mixture is fragrant and the fat takes on a reddish hue, about 1 minute. Drizzle the mixture over the chicken and serve.
