Butterfly the chicken breasts if greater than 3/4 inch thick.
Mix flour, salt and pepper in a big dinner plate.
Beat one egg in a shallow bowl.
Dip chicken into egg. Turn to coat the other side.
Place chicken into flour mixture. Press chicken evenly to coat the entire side. Flip chicken to coat the other side. Shake excess flour off chicken and set aside. Repeat for remaining breasts.
Heat a non-stick pan to medium heat. Allow 3-5 minutes. Then add olive oil and butter. Once the butter is melted, add chicken to the pan. You should hear a distinct “TISSS” sizzle.
Leave the chicken alone for about 5 minutes. Then flip and do the same. Be sure not to crowd the pan. To best crisp the chicken, it needs room in the pan. I used a 12-inch pan and worked in two batches.
Add garlic as the chicken nears completion (about 15 minutes in)
You can finish the chicken in the oven. The breasts need to reach 165 degrees. Either way, once the chicken is done, set aside on a clean plate or platter.
In the pan you cooked the chicken, add mushrooms and shallot (or onion). Cook for 3-5 minutes/until mushrooms and shallot soften.
Add white wine, lemon juice, capers and chicken stock. If you splurged on fresh thyme, add it now. Scrape the bottom of the pain as the liquids bubble up.
Continue cooking the sauce over medium heat. The sauce will eventually thicken. Continue intermittently scraping the bottom of the pan.
Once the sauce is thick, plate the chicken and spoon over the sauce.