We just had this for lunch and it was really easy to make and very yummy! I did cheat a bit and used canned peaches instead of fresh ones, store-bought chicken broth instead of homemade one, but... both kids finished everything on their plates, with no fuss. I even got a "Yummy, Mommy" from my 2-yr old. 
Here it is:
Sauteed Chicken Breasts with Peaches
4 skinless chicken breasts (about 6oz each), trimmed of excess fat
Salt and freshly ground black pepper
Unbleached all-purpose flour
2 tbsp. unsalted butter
? cup thinly sliced shallots
? cup balsamic vinegar
? cup chicken broth, preferably homemade
? cup heavy cream
2 large ripe, but firm peaches (about 1 lb. total), cut into ?-inch-thick wedges
2 tbsp. shredded fresh basil, or chopped fresh thyme leaves
1. Remove fillets (the finger-size muscle on the underside of each breast) and set aside. Using the flat end of a meat pounder, lightly flatten each breast.
2. Season the chicken breasts and fillets with salt and pepper, then dredge through the flour, shaking off any excess.
3. Melt 1 tbsp. of butter in a large skillet over medium-high heat. Add the chicken and saut? until just cooked through, about 2 minutes per side. To test for doneness, cut into the thickest part of the chicken and see. Flesh should be opaque, no trace of pink.
4. Transfer chicken to a plate and cover while you make the sauce.
5. Add remaining 1 tbsp. butter to skillet. Add shallots and saut? until tender, about 30 seconds. Add balsamic vinegar and cook until reduced by half, forming a thick glaze (about 2 minutes).
6. Add chicken broth and cream, + any juice on the chicken plate. Cook until sauce is reduced by half and lightly coats the back of a spoon, about 2 minutes.
7. Add peaches and toss until just warmed through. Do not overcook. Remove from heat and stir in basil. Adjust seasoning with salt and pepper, if necessary.
8. Place chicken on platter. Spoon sauce and peaches over chicken and serve immediately.
Here it is:
Sauteed Chicken Breasts with Peaches
4 skinless chicken breasts (about 6oz each), trimmed of excess fat
Salt and freshly ground black pepper
Unbleached all-purpose flour
2 tbsp. unsalted butter
? cup thinly sliced shallots
? cup balsamic vinegar
? cup chicken broth, preferably homemade
? cup heavy cream
2 large ripe, but firm peaches (about 1 lb. total), cut into ?-inch-thick wedges
2 tbsp. shredded fresh basil, or chopped fresh thyme leaves
1. Remove fillets (the finger-size muscle on the underside of each breast) and set aside. Using the flat end of a meat pounder, lightly flatten each breast.
2. Season the chicken breasts and fillets with salt and pepper, then dredge through the flour, shaking off any excess.
3. Melt 1 tbsp. of butter in a large skillet over medium-high heat. Add the chicken and saut? until just cooked through, about 2 minutes per side. To test for doneness, cut into the thickest part of the chicken and see. Flesh should be opaque, no trace of pink.
4. Transfer chicken to a plate and cover while you make the sauce.
5. Add remaining 1 tbsp. butter to skillet. Add shallots and saut? until tender, about 30 seconds. Add balsamic vinegar and cook until reduced by half, forming a thick glaze (about 2 minutes).
6. Add chicken broth and cream, + any juice on the chicken plate. Cook until sauce is reduced by half and lightly coats the back of a spoon, about 2 minutes.
7. Add peaches and toss until just warmed through. Do not overcook. Remove from heat and stir in basil. Adjust seasoning with salt and pepper, if necessary.
8. Place chicken on platter. Spoon sauce and peaches over chicken and serve immediately.
