Place the potatoes in a large pot and cover with cold water. Add 1/2 teaspoon salt and bring to a boil. Cook until the potatoes are fork-tender, about 15 minutes.
While the potatoes cook, season the chicken breasts with the remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Heat the olive oil in a large skillet over medium heat. Add the chicken and cook until golden on both sides and cooked through, about 6 minutes per side. Remove the chicken to a plate.
Add the garlic to the same skillet and cook for 30 seconds until fragrant. Pour in the chicken broth and scrape up any browned bits. Stir in the Dijon mustard and heavy cream. Simmer the sauce for 3 to 4 minutes until it thickens slightly. Return the chicken to the skillet and spoon sauce over the top.
Drain the potatoes and return them to the pot. Add the butter, milk, and remaining 1/4 teaspoon pepper. Mash until smooth, then stir in the parsley and chives.