Season the Chicken: Pat chicken thighs dry and season well with salt and pepper.
Sear the Chicken: Melt butter and oil in a large skillet, then cook the chicken for about 5 minutes until browned.
Cook the Other Side: Flip the chicken and cook for about 2 more minutes.
Add Shallots & Simmer: Scatter sliced shallots over the chicken, cover, lower heat, and cook 8–9 minutes until chicken reaches 165°F (74°C).
Remove Chicken: Transfer cooked chicken to a plate and keep warm.
Make the Sauce: Stir cream and Boursin into the pan drippings, increase heat, and cook until melted and bubbly, about 5 minutes.
Serve: Pour the creamy sauce over the chicken and garnish with chives if desired.