The mushrooms should go in first and stay there until they lose their moisture and begin to color. Pale, steamed mushrooms don’t contribute much. Properly browned mushrooms give the sauce depth.
I try to use crimini because they give the sauce better color and more flavor than white button mushrooms. Shiitake are always a good choice and bring a deeper savory quality without taking over the dish. Morels give it a particularly woodsy French hunter character. Oyster mushrooms, maitake and a few dried porcini also work well.
I generally use a New Zealand Sauvignon Blanc. Its acidity cuts through the chicken fat and keeps the sauce lively. You should never cook with a wine you wouldn’t drink. It doesn’t need to be expensive, but reducing bad wine only produces a more concentrated version of bad wine.
Then there is the cognac. Flaming it is entirely optional, but it makes dinner more entertaining. We have an induction stove, so I use a small butane kitchen torch. Take the pan off the heat, add measured cognac and ignite the surface carefully. Keep your face, hair and sleeves well back, and never pour cognac directly from the bottle into a hot skillet.
You can also simply let it simmer for a minute. The chicken won’t know it missed the show.
Once the sauce is underway, return the thighs to the skillet skin-side up and keep the liquid below the browned skin. In my skillet, they usually take about 30 minutes to reach 165°F.
I’m a great believer in thermometers. I use three when I cook a steak, which may be excessive, but it beats cutting into the meat and guessing. Chicken thighs vary in size, skillets vary and stove settings are never quite as exact as we pretend.
Use the clock as a guide and the thermometer as the answer.
The sauce can go in several directions. Reduce it and swirl in cold butter for a light, glossy finish. Add cream if you want something richer. If you’re serving it with pasta, save some pasta water and use it to help the sauce cling to the noodles.
Flour works too, but don’t dump it directly into the skillet. Pépin’s trick is to put it in a fine-mesh strainer and shake a light dusting over the chicken or sauce while it cooks. It disperses gradually and is much less likely to clump.
For a darker brown sauce, I sometimes add a few drops of Gravy Master near the end. Start small. It works quickly.
Serve the chicken with mashed potatoes, noodles, rice, polenta or good bread. The bread may be the most important part, because leaving this sauce behind would be a mistake.