Thoroughly pat chicken thighs dry with paper towels. Season generously with salt on both sides.
Sear for Crispy Skin
Heat olive oil and 2 Tbsp of butter in a large skillet over medium-high heat. Once the butter is melted, place chicken thighs skin side down and cook without moving for 6-8 minutes, until skin is deeply golden brown and crispy. (The chicken is ready to flip when it easily releases from the pan.)
Transfer the chicken to a wire-rimmed rack sprayed with oil. Place in oven and cook for 20 minutes.
Make the Champagne Sauce
Turn the heat off from under the pan the chicken cooked in. Remove some of the chicken fat until there is just about 2 tablespoons left in the pan. Turn the heat to medium and add the shallots.
Cook 2 minutes until softened. Add the garlic and saute until fragrant, about 1 minute.
Slowly pour in the champagne to deglaze, scraping the browned bits off the bottom. Allow the sauce to come to a simmer and reduce the liquid by half, allowing to simmer for about 5 minutes.
Stir in the heavy cream and Dijon mustard; simmer until slightly thickened.
Remove chicken from the oven and place on top of the champagne sauce. Serve!