Start the Sauce
In a sauce pot, over medium heat, add the white wine, white wine vinegar (or lemon juice) and shallots. Bring these to a boil.
1 C Dry White Wine, 2 Tbsp White Wine Vinegar, 2 Tbsp Shallots, Finely Chopped
Let this mixture reduce until about 2 Tbsp of liquid remains, stirring occasionally. It should take around 7-10 minutes.
Cook the Chicken
Heat the olive oil in a large pan or skillet over medium-high heat.
2 Tbsp Olive Oil
While that heats up, season both sides of the chicken thighs with a few heavy pinches of salt and pepper.
4 Boneless Skinless Chicken Thighs, Salt and Pepper to Taste
Add the chicken to the pan and cook for 5-6 minutes on each side, until the internal temperature reaches 165 degrees F and the juices run clear.
Let the cooked chicken rest while you finish the sauce. You can tent it with foil if you like.
Finish the Beurre Blanc
Once the white wine mixture has reduced, drop the heat to low.
Stirring constantly, add in 2 Tbsp of your cold butter. Stir and cook until the butter is almost completely melted. Then add another 2 Tbsp.
2 Sticks Cold Unsalted Butter, Divided into 1 Tbsp Pieces
Repeat the previous step until all of the butter is mixed into the sauce. Do not make this sauce over high heat. The goal is not to completely melt the butter, but to emulsify (form a semi-stable mixture) it into the sauce. Using too much heat will break the emulsification and leave you with flavored melted butter - still great, but it won't stick to the chicken very well. You should end up with a gravy-like consistency if you use low heat and emulsify properly. As you stir the butter into the sauce, it should start to thicken and become cloudy. Right before the last piece of butter finishes dissolving into the sauce, cut the heat.
Add a heavy pinch of salt and pepper, and then taste to check seasoning.
Serve
Serve the dish right away. Add the chicken to some rice or mashed potatoes and top with the beurre blanc sauce.