Coat the bottom of a skillet generously with olive oil and heat over medium-high heat.
Set up your dredging station. Place your beaten egg in one dish, your flour in another and your panko in another.
Dredge your chicken (both sides!) first in the flour, then the egg (making sure to coat the chicken very well in the egg) and last in the seasoned panko crumbs. Make sure to thoroughly coat the chicken in the crumbs.
When the oil is very hot (a droplet of water will dance on top when it's ready), add your chicken to the pan.
Cook the chicken until golden brown on one side - about 3-4 minutes - then flip and cook on the other side for 3-4 minutes, until browned.
Remove the chicken from the pan. if you still have tons of oil in the pan, drain some of it so that there's about 1 tbsp leftover in the hot pan.
Turn the heat down to medium and add the garlic. Move it around so that it doesn't burn.
Add a few cubes of butter and swish the pan around - avoiding too much direct heat so that the butter and garlic don't burn - and let the butter melt completely.
Add the white wine. Leave the pan on the heat and let the liquid reduce by about half.
Add the rest of the butter and stir it in.
Once the butter is melted, add about 75% of the chicken stock. Let the stock simmer and thicken slightly, about 2 minutes.
Add the capers and a pinch of salt. Stir.
Taste the sauce. If you need to add more salt, add it and stir it in. If you need to add the rest of the stock, add it. I typically add the rest of the stock, but if you prefer a more buttery sauce, you don't need it.
Tuck the chicken into the sauce and let it simmer for 2-3 minutes. Serve hot with a lemon wedge over roasted potatoes and veggies, if desired.