Score the fat on top of the duck breast diagonally in both directions. Make sure to cut all the way through the fat without cutting into the meat. This will help the fat render properly. Season generously with salt and pepper.
Heat a saute pan over medium heat for several minutes and then place the duck in the pan, fat-side down.
Let the fat render over medium heat for several minutes (5-10 depending on the size) until it's dark brown and crispy. Drain off some of the duck fat periodically so the meat isn't swimming in fat (save in the fridge for later use).
Flip the breast and allow it to cook on the opposite side for a few minutes. You're aiming for a medium rare breast with crispy (not chewy) fat. I like to keep a thermometer on hand. Remove from the heat when the center is around 125 degrees (it will continue to cook). Allow the meat to rest for at least 5 minutes.
While the duck is resting, prepare the pan sauce. Drain any remaining fat from the pan and then deglaze with red wine. Use a spatula to scrape up any brown bits from the bottom and incorporate them. Once the wine is almost cooked away, add the shallots and a sprinkle of salt. Allow them to cook for 2 minutes, then add the mushrooms. Brown the mushrooms for 2-3 minutes and then add the chicken stock.
Allow the stock to reduce significantly until it's a thin sauce consistency. Remove the pan from the heat and stir in the butter immediately.
Slice duck on the bias into thin strips. (If you accidentally undercook the meat you can briefly stick it in a 375 degree F oven. Be very careful not to overcook in this scenario.)