I start by patting the chicken thighs completely dry with a paper towel, since any extra moisture will stop them from browning properly. Then I season both sides generously with salt and pepper. In a large skillet, I heat the olive oil and butter together over medium-high heat until it shimmers and starts to bubble slightly.
Once the pan is hot, I lay the chicken thighs in without crowding them, giving each piece room to brown. I let them sear for about 5 to 6 minutes per side, resisting the urge to move them around too soon. When they're golden on both sides and nearly cooked through, I transfer them to a plate and set them aside.
In the same skillet, I add the chopped onion and let it soften for a couple of minutes, scraping up any browned bits left behind from the chicken. Those bits are packed with flavor, so I never skip this part. Then I stir in the minced garlic and cook it for about 30 seconds, just until it turns fragrant.
I pour in the chicken broth first and let it simmer briefly before whisking in the heavy cream and Dijon mustard. I keep whisking until the sauce looks smooth and slightly thickened, which usually takes a couple of minutes over medium heat. This is also when I add the fresh thyme for extra depth.
Once the sauce looks right, I return the chicken thighs to the skillet along with any juices that collected on the plate. I let everything simmer together for 8 to 10 minutes, spooning the sauce over the chicken occasionally so it stays coated and the flavors blend.
Right before serving, I check that the chicken has reached an internal temperature of 165°F, then sprinkle the fresh parsley over the top. I like to let the dish rest for a minute or two off the heat, which helps the sauce cling to the chicken a little better.