Preheat oven to 400°F (200°C) if you plan to finish the breast in the oven (I do this when the thickest part needs a few more minutes).

Season both sides of the chicken breasts generously with salt and pepper. Let them rest for 5 minutes.

In a heavy oven-proof pan (I use a 10-inch skillet), heat the vegetable oil over medium-high heat until it’s shimmering.

Place the chicken breasts skin-side down without moving them. Let the skin crisp and turn a deep golden brown—about 5-6 minutes from my experience. If it’s not browning, increase the heat slightly but don’t burn the skin.

Flip the breasts so the skin side is up. If you’re finishing in the oven, transfer the pan now and roast for another 8-10 minutes or until an instant-read thermometer inserted into the thickest part reads ~155-160°F (they will carry over-cook to 165°F after resting). If you’re finishing on the stovetop only, reduce heat to medium and cook for another 7-10 minutes, monitoring temperature and doneness.

Remove the chicken from the pan and let it rest on a cutting board while you make the sauce (about 5 minutes).

With the pan back on the heat, pour off most of the fat from the skin (but leave about 1 tablespoon). Add the minced shallots and sauté for ~1 minute until fragrant (but not browned).

Pour in the white wine and use a wooden spoon to scrape up the browned bits from the pan (fond). Let the wine reduce by about half (1-2 minutes).

Add the chicken broth, Dijon mustard, and fresh thyme. Bring to a simmer and let the sauce reduce until it has thickened slightly (about 3-4 minutes) and you can see it coat the spoon.

Remove from heat, whisk in the cubes of cold butter one at a time until melted and the sauce is smooth and glossy. Squeeze in the lemon juice and stir in chopped parsley. Taste and adjust salt & pepper.

Slice the resting chicken breasts (if you like, at a slight angle) and place them on plates. Spoon the warm sauce over the breasts, including some pan juices. Serve immediately.