Rinse the chicken and pat dry. Put on a large plate and refrigerate, uncovered, at least 4 hours or overnight to dry out the skin.
Pat the chicken dry and season the cavity with salt and pepper. Put the chicken on a rack in a roasting pan, tucking the wings underneath. Preheat the oven to 450 degrees F.
Cook the butter, garlic and walnuts in a small skillet over low heat until the butter is melted and the nuts are lightly toasted, 5 to 7 minutes. Brush the chicken with 2 tablespoons of the garlic butter; transfer the remaining butter mixture (including the garlic and nuts) to a large bowl. Add the parsley, thyme and bread and toss to coat. Fill the cavity of the chicken with about two-thirds of the stuffing. Tuck the remaining stuffing on the baking sheet underneath the chicken. Sprinkle the chicken with 2 teaspoons salt and 1/4 teaspoon pepper.
Transfer the roasting pan to the oven and roast until the stuffing is toasted, about 20 minutes. Add 2 cups chicken broth to the pan and roast 40 more minutes. Use a spoon to scoop the stuffing out of the chicken and transfer to the bottom of the roasting pan with the rest of the stuffing. Reduce the heat to 425 degrees F and continue roasting until an instant-read thermometer registers 165 degrees F, 30 to 40 minutes.
Transfer the chicken to a platter and let rest 15 minutes. Transfer the stuffing to a bowl. Put the roasting pan on the stovetop over medium-high heat. Add the remaining 2 cups broth to the pan and cook, scraping up any browned bits with a wooden spoon, until almost syrupy, 5 to 8 minutes. Return the stuffing to the pan to soak up the juices. Serve with the chicken.