Chicken Cacciatore

You can use chicken breasts if you like, but much cheaper and more likely to appear in a household in Italy would be a whole chicken, cut up:

Turn the chicken with its back up.  With shears, cut on either side of the backbone and remove it for stock.  You can do this as you are doing the rest of the recipe, also adding the wing tips and any other waste pieces of the chicken, neck gizzard heart, salt and pepper but not the liver.  Just simmer in a saucepan of water, adding onion, carrot and celery if you like.    Skim off any foam that rises in the first few minutes of cooking.

Cut off the leg quarter by just cutting through the skin and fold the thigh and leg outward.  separate the thigh from the chicken.  Find the joint between the leg and thigh(there will be a line of fat to follow if you can see it). cut off the leg at the joint.
Fold the wing outward as well, and cut through the joint.  take a little of the breast meat with it if you want this person to get a bit more meat.  
Find and cut out the wishbone.  This is a little fiddly.
Lay the rest of the chicken flat.  You might cut through the ribs that have little meat on them, with the shears and add to the stock.
Flatten the breast out, inside of the carcass up.  Push it down hard then split the breastbone with a large knife, or cut on both sides of the keel and add to the stock.  Cut each breast half across the narrow dimension, but move toward the heavier end of the breast so the portions are near equal.

Lightly flour the pieces by dropping them one by one into a plastic bag full of flour with a little salt and pepper in it.  Remove and lay into a frying pan coated with olive oil (Not extra virgin)  allow to brown lightly on all sides, then remove to a platter. 

Pour out accumulated oil and fat.

Chop a couple of onions very coarsely(perhaps 8ths) and drop into the oil.  Keep it moving as it fries.  Add celery chunks and carrot chunks into the pan and continue to brown.  Add several crushed cloves of garlic.  Return the chicken to the pan and mound in quartered mushrooms and plenty of coarsely chopped peppers of your choice.  Keep the green ones in a small minority if you use them at all.
Add fresh or dried rosemary to taste
Add a couple of cups of your hot broth(after straining), and a couple of cups of white wine and a can of crushed tomatoes(or fresh skinned and chopped).  Salt and pepper to taste and cover  and simmer over low heat, till all is cooked through stirring occasionally.  Alternatively, cover and place in a 325 degree oven till all is cooked through.   In both cases, leave uncovered if it appears juicier than necessary.