Pat chicken pieces dry with a paper towel, then season generously on both sides with kosher salt and black pepper. If not already at room temperature, allow the chicken to sit out.
Heat unsalted butter in a large skillet over medium-high heat. Add the seasoned chicken pieces skin side down, then cook until the skin is golden brown. Flip the chicken over and cook.
Add shallot, fresh thyme sprigs, and bay leaf, then pour inof white wine. Reduce the heat to medium low, cover the pan with a lid, then cook until the chicken is fully cooked and has reached an internal temperature of 165F.
Strain the liquid through a fine mesh sieve set over a large measuring cup or bowl, then pour the liquid back into the pan. Turn off the heat, then transfer cup of the sauce back to the mixing bowl or measuring cup. To this, whisk in large egg yolks until tempered. Pour the egg and sauce mixture back into the pot, then stir.
Serve the chicken hot with plenty of the creamy mustard sauce poured on top. Finish with a pinch of flaky sea salt and fresh thyme leaves. Enjoy!