Preheat the oven to 400 degrees. Bake the potatoes until soft, about 1 hour.
Meanwhile, in a large soup pot, brown the bacon over medium heat. Add the carrots, celery and onions. Season with salt and pepper and cook until the vegetables are translucent, 3 to 5 minutes. Add the chicken, thyme and broth; bring to a simmer and skim any foam that rises to the top. Reduce the heat and barely simmer until the chicken is cooked through, about 40 minutes. Discard the thyme. Remove the chicken and let cool. Pull off the meat and shred into bite-sized pieces; cover and set aside. Season the broth with salt and pepper.
Melt the butter in a skillet over medium heat. Add the breadcrumbs and cook, stirring, until toasted, about 3 minutes; set aside. Halve the baked potatoes and scoop the flesh into a bowl; mash with a fork and season with salt and pepper. Sift the flour over the potatoes and add the toasted crumbs, parmesan and nutmeg; lightly knead to make a firm dough. Roll into 1/2-inch-wide cylinders; cut into 1/2-inch pieces and round the edges to make oval-shaped gnocchi. Bring the soup to a boil and drop in the gnocchi. Cook until they float to the top, 1 to 2 minutes; transfer the gnocchi to warm bowls with a slotted spoon.
Beat the eggs with a fork; use the fork to "paint" ribbons of egg across the soup. Let set, about 30 seconds. Stir in the chicken meat, parsley, basil and spinach; season with salt and pepper. Ladle the soup over the gnocchi. Serve with lemon wedges and more Parmesan.
Photograph by Yunhee Kim
Recipe courtesy Alexandra Guarnaschelli for Food Network Magazine