Sprinkle chicken thighs with kosher salt and black pepper. Place chicken, skin side down, in a large cast-iron skillet over medium. Cook until fat is rendered and skin is crispy, about 15 minutes. Flip thighs, and cook until a thermometer inserted into thickest portion of meat registers 165°F, about 10 minutes.
Cut off and discard ends of lemon. Cut lemon into quarters; remove and discard seeds. Transfer lemon to a food processor or blender, and process until finely chopped, about 1 minute. Add oil, herbs, vinegar, mustard, and garlic; process until mostly smooth and well combined, 30 seconds to 1 minute. Transfer vinaigrette to a small bowl, and whisk in salt and pepper.
Transfer chicken to a platter; drizzle with about ⅓ cup vinaigrette. Reserve remaining vinaigrette for another use.
