Preheat the oven to 400 degrees F (200 degrees C).
Melt the Unsalted Butter in a large, heavy-bottomed oven-safe pot or skillet set over medium-high heat. When the butter foams, add the Shallot to the pot, season with Kosher Salt, and sauté them in the butter until they begin to soften and caramelize, approximately 10-12 minutes. Lower the heat as needed if shallots are browning too quickly.
Add the Dijon Mustard and Tarragon and stir until the mustard coats the shallots evenly and begins to brown.
Add the White Wine and bring to a simmer, scraping the bottom of the pot with a wooden spoon to release any brown bits.
Meanwhile, season the Bone-in, Skin-on Chicken Thighs and Drumsticks generously with Kosher Salt and Freshly Ground Black Pepper on both sides. When the wine simmers, add the chicken pieces to the pan and transfer the pan to the oven. Cook for 40-50 minutes, or until the skin is nicely brown and crisp. Remove pan from oven.
Using a spoon, swirl around the sauce and take a taste. If it tastes good, serve it. If it tastes fatty, transfer the meat and shallots to a platter, pour the juices into a Pyrex and let them sit until the fat rises, about 5 minutes. Skim off the fat.
Return the juices, chicken and shallots to the pan. Return to the oven or bring to a simmer stove top to reheat. If chicken isn't as brown as you would like, you can stick it under the broiler too. Serve immediately with crusty bread and a salad.