Pat chicken dry with paper towels. Sprinkle evenly on all sides with 2 teaspoons of the pepper and 1 teaspoon of the salt, pressing gently to adhere. Place chicken skin side down in a cold 12-inch skillet. Cook over medium, undisturbed, until skin is golden brown and very crisp, about 15 minutes. Turn chicken skin side up; cook, undisturbed, until a thermometer inserted into thickest portion of chicken registers 165°F, about 10 minutes. Transfer chicken to a plate and set aside, reserving drippings in skillet.

Add shallot, garlic, thyme, remaining 1 teaspoon pepper, and ¼ teaspoon salt to drippings in skillet. Cook over medium, stirring often, until shallots are softened, 2 to 3 minutes. Remove from heat, and add Cognac. Return skillet to medium; cook, stirring constantly to scrape up any browned bits on bottom of skillet, until Cognac is nearly evaporated, about 30 seconds. Pour in broth, and bring to a simmer over medium. Simmer, stirring occasionally, until slightly reduced, about 2 minutes. Whisk in crème fraîche and Dijon mustard until smooth. Reduce heat to medium-low; simmer, whisking often, until sauce is thick enough to coat back of a spoon, about 5 minutes.

Return chicken thighs to skillet, skin side up, and simmer over medium-low, swirling skillet occasionally, until chicken is heated through, about 1 minute. Transfer chicken to a serving plate and spoon sauce over top; garnish with chives and additional pepper. Serve immediately.