Season chicken all over with kosher salt and pepper, transfer to a large plate, and refrigerate uncovered for at least 30 minutes and up to 4 hours. In a large skillet or Dutch oven, heat 1 tablespoon butter over medium-high heat until melted. Sear chicken in skillet, turning once, until browned on both sides, about 3 minutes per side. Transfer chicken to a plate. Do not wipe out skillet.

In the now-empty skillet, add 2 tablespoons butter and heat over medium-high until melted. Add mushrooms and 1 teaspoon kosher salt and cook, covered, until mushrooms have released their liquid, about 5 minutes. Uncover and continue to cook, stirring frequently, until mushrooms begin to brown, about 6 minutes. Add shallots, thyme, and smoked paprika or Aleppo pepper, and cook, stirring frequently until shallots are softened, about 3 minutes. Add flour and cook until it smells slightly toasty, about 1 minute.

Add wine, and cook until wine is reduced by half, about 3 minutes. Add broth, fish sauce, and reserved chicken, and bring to a simmer. Cover and cook over medium heat, flipping chicken halfway through, until the chicken registers 160°F (71°C) with a digital thermometer, about 5 minutes. Transfer the chicken to a serving dish, leaving mushrooms and sauce in skillet.

Increase the heat to medium-high and simmer sauce, uncovered, stirring occasionally until slightly reduced, about 5 minutes. Whisk in heavy cream and Dijon mustard and continue to simmer, whisking rapidly, until thickened to a velvety consistency that coats the back of a spoon, about 3 minutes. Remove skillet from heat and stir in minced chives or tarragon. Season to taste with salt and pepper. Nestle chicken back into sauce, and flip to coat in sauce and to warm through. Serve chicken with sauce immediately.