For the chicken: Coat a large saute pan with olive oil, toss in the shallots, season with salt and bring the pan to a medium-high heat. Cook the shallots for 3 to 4 minutes, stirring frequently.
Toss the chicken livers into the pan, stir to combine and cook for 2 minutes. Toss in the mushrooms, stir to combine and season with salt. Cook the mushrooms until they are soft and wilted, 4 to 5 minutes.
Add the wine and cook, stirring frequently, until the mushroom mixture is almost dry. Remove from the heat and let cool.
Transfer the cooled mushroom mixture to a medium bowl, add the goat cheese and chives and mix until it is a homogeneous mixture. Put the mushroom mixture in a disposable piping bag and reserve.
Using a paring knife or boning knife, make an incision in the thick end of each chicken breast. Insert the knife all the way into the chicken breast and slide it back and forth to create a pocket. Try not to poke the knife through the top or bottom of the chicken.
Squeeze the mushroom filling into each chicken breast.
In 3 separate wide, shallow containers, set up a standard breading procedure: one container of flour, one with the beaten egg mixture and one with the panko.
Season the chicken breasts with salt. Using one hand for wet things and one hand for dry things, dredge each chicken breast in the flour. Dip in the egg wash and then pack panko onto each chicken breast. Lay the chicken breasts on a sheet tray and let them rest in the refrigerator for at least 1 hour.
Preheat the oven to 300 degrees F. Set up a tray lined with paper towels. Take the chicken breasts out of the refrigerator and let them come to room temperature for 10 to 15 minutes.
Coat a large saute pan with olive oil and bring the pan to a medium-high heat. Work in batches if all the chicken breasts do not fit in the pan at the same time. Cook the chicken until well browned on both sides, 5 to 6 minutes per side. Remove the browned chicken and blot it on the paper towels; season with salt immediately. Repeat with the remaining chicken.
When all the chicken has been browned and blotted, place the chicken breasts on a wire rack set over a sheet tray and continue to cook for 10 to 15 minutes more in the oven.
For the sauce: Meanwhile, ditch the fat from the pan and add the wine. Reduce the wine by half. Add half the chicken stock and reduce that by half. Add the remaining chicken stock and reduce that by half also (this will take a little time so be patient).
Taste the reduced stock and season with salt if needed. Remove the pan from the heat and swirl in the butter.
Plate the chicken and spoon the pan sauce over the chicken.