Bring a small pot of water to a boil, then reduce to low and add the dried peppers. Cover the pot and let them simmer for about 30 minutes. Then turn the heat off, keep the peppers covered and let them sit until they're soft. Another 30 minutes is sufficient, but the longer the better. When they're done, remove them from the pot but SAVE the cooking liquid! Remove the stems from the peppers and use your hands to open them and remove the seeds. (You can also use a small spoon or the back of a knife to gently scrape them off.) Set the prepared peppers aside.
While the peppers are hydrating, preheat the oven to 375°F, adjust a rack to the center.
Then coat the bottom of a large, deep, oven proof sauté pan with olive oil. Place it over medium heat, add the onions and garlic and sauté, stirring periodically, until the onions are caramelized and soft, about 15 minutes.
Add the fresh tomatoes to a blender or food processor with the hydrated, seeded peppers and 3/4 cup of the cooking liquid. (Now you may discard the rest of the liquid.) Blend until smooth, and then pour this into the sauté pan with the onion mixture. Add the balsamic vinegar and stir to blend. Season with a bit of salt and pepper -- and then season with more (if necessary) to taste after the chicken is done. (Here's How to Season to Taste.)
Place the pan in the preheated 375°F oven and braise just until the chicken is cooked through, about 40 minutes. At the halfway point, cover the pan lightly with foil. Try to remove the pan as soon as the chicken isn't pink anymore -- keep checking towards the end of the cooking time. (You can use a sharp knife to poke into the thickest part of one of the breasts to see if it's done.)
Let the chicken "rest" for a few minutes before serving. It's delicious over rice, pasta, or a nice chunk of rustic bread.