Heat the olive oil in a large skillet over medium heat. Season both sides of the chicken breasts with salt and pepper.
Add the chicken to the skillet and cook for 5–6 minutes per side, until golden brown and cooked through. Transfer the chicken to a plate and set aside.
Pour the chicken broth into the same skillet, using a wooden spoon to scrape up any browned bits from the bottom.
Reduce the heat to medium-low. Stir in the sour cream, Dijon mustard, and fresh dill until the sauce is smooth and well combined.
Let the sauce simmer gently for 2–3 minutes, stirring occasionally, until slightly thickened.
Return the chicken to the skillet and spoon the sauce over the top. Cook for 2 minutes, just until the chicken is warmed through.
Garnish with additional fresh dill and serve warm.