Preheat the oven to 375°F / 190°C / Gas Mark 5, no fan (if you have a convection oven reduce the temperature by about 20° or follow the manufacturer's instructions).
Create a pocket for the cheese: Place the chicken breast on a cutting board. Press a hand on top of the breast and carefully insert the tip of a knife into the thickest part of the chicken breast. Cut 3/4 of the way through the breast to create a pocket without cutting all the way through. Season the chicken with salt and pepper on all sides, stuff the pocket with the cheese. Wrap each breast in prosciutto ham, use as many slices as it's needed to completely cover the meat.
Heat the oil in an oven-proof pan. When hot add the chicken, cook on both sides over high heat until nicely browned (this should take about 2 minutes per side).
Move the chicken to the side of the pan and add the butter. When it's bubbling add the pressed garlic and cook, stirring, for about 30 seconds.
Add the wine and cut into 8 wedges peaches (peeled or skin-on), cook for about 1 minute, scraping all the brown bits from the bottom of the pan.
Place the pan in the oven (if you don't have an oven-proof pan, just transfer the content of the pan into a baking dish) and bake until the chicken is cooked through. It should take about 13-15 minutes for medium chicken breasts ad 15-18 minutes for large chicken breasts. It can vary, depending on how long you've cooked the chicken in the pan and how large your chicken breasts are. To be sure if the chicken is cooked through, it's best to check its internal temperature with a meat thermometer - stick it into the thickest part of the breast. The temperature should be 160°F / 70°C (the safe internal temperature is 165°F/74°C, but since we're letting the meat rest for a while, the residual heat will cook the chicken).
Transfer the chicken to a plate and leave to rest for 5-10 minutes before cutting it into slices.
Cook the sauce that has remained in the pan for a couple of minutes until thickened, then strain through a sieve. Pour the sauce over the chicken.
Now you can: make the balsamic reduction or use a store-bought balsamic glaze or add about 1 teaspoon balsamic vinegar to the sauce or leave the dish as is. I prefer to serve it drizzled with homemade balsamic reduction.
How to make the balsamic reduction: add the balsamic vinegar into a very small pot and cook over low heat until reduced in volume, about a half, and thickened (don't wait until the sauce will be very thick as it will thicken more while cooling). Take the pot off the heat and let it cool for about 5 minutes. When it's thickened, drizzle it over the chicken.