Put vegetables in pot and cover with water and wine. If you want, you can also add an inch or two of fresh ginger, a Thai chile or a stalk of lemongrass. Bring to simmer. Cook until vegetables taste like nothing and broth tastes like everything. Salt as you go.
Meanwhile, remove chicken skin. When broth is done, remove vegetables and add chicken. Simmer, lid on, until done- about 20 minutes. Under no circumstances should your broth be boiling while chicken is in it. Save cooked chicken meat for part two (or use half here, half in part two).
While chicken cooks, place skin in a cold, non-stick pan and turn heat to medium. Fry on both sides until golden brown and reserve for later use. Use rendered chicken fat to brush the tops of the bread. Toast bread. Crack eggs into remaining fat (you may have to add a little extra oil). When whites are set (30 seconds in) add enough broth to nearly cover the eggs. Simmer until eggs are cooked to your liking.
Place toasted bread in shallow bowls and fill with broth. Top with egg, herbs, olive oil, salt and pepper and pecorino or parmesan cheese.