In one, add and mix the flour, salt, and pepper; in another, place the beaten eggs; and in the third, add and mix the grated cheese and breadcrumbs.
Heat a large frying pan over medium heat.
When hot, add the olive oil and, working quickly, dip a chicken piece in the flour, then the egg, and then in cheese-breadcrumbs, and add to the pan.
Repeat with as many pieces as will fit without crowding; if necessary, cook in two batches.
Sauté until the undersides are brown and crispy, about 5 minutes, then flip and continue to sauté until cooked through, about another 3 minutes or until an instant-read thermometer reads 165°F.
Add the mushrooms to the pan and sauté lightly.
If you cooked the chicken in batches, return them all to the pan.
Add the tomatoes with all their juice, the olives, basil, capers, and brine and bring to a low boil.
Cook for 5 minutes, using a spatula to scrape up any fond on the bottom of the pan.
Drizzle on the vinegar and serve. Pass more vinegar and olive oil for drizzling.