Begin by trimming the chicken breasts if they need trimming. I use my beloved kitchen scissors and snip off any shaggy bits of fat or gristle.
Pat them dry with paper towels (to helps them brown) and sprinkle them with some kosher salt.
Heat a very large pan over medium-high heat and add the butter, which should melt and foam, and the olive oil, which should stop the butter from burning. When the fat is all very hot but not turning black (recipes say "when the foam subsides"), add the chicken breasts in one layer. Now leave them alone for 5 or so minutes while the bottom gets nice and crusty and brown.
Use tongs or a spatula to flip them over as they brown, and then cook another 4 or so minutes until the bottom is very brown and the chicken is cooked through. You may want to cut a piece open to check, but pressing a cooked breast with your fingertip will give you a feel for its doneness, which will develop over time into the skill of knowing when it's done without cutting it.
Pile the chicken on a heat-proof plate and pop it into a 200ºF heated oven while you "deglaze" the pan. Add the garlic, onion and mushrooms until tender. Pour half the broth in and scrape with a spatula to dissolve all the yummy browned bits while the broth bubbles and boils furiously in the hot pan.
When the wine has cooked down about halfway, add the broth & juice, turn the heat to high, and cook until the sauce seems syrupy and delicious. Taste it for salt, then drizzle it over the chicken, sprinkle with parsley, and serve. Sometimes, if the chicken has browned but doesn't seem quite cooked through, I leave it in the pan while I make the sauce so that it can simmer in there a bit longer.