Place another skillet over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Season the chicken breasts with salt and pepper and add them to the pan with the oil. Cook the chicken in the skillet until golden brown and cooked through, 3-4 minutes per side.
Remove the cooked chicken from the pan and add the remaining 2 tablespoons butter. When the butter has melted, sprinkle in the flour and cook for about 1 minute. Slowly whisk the chicken stock, cream and mustard into the butter-flour mixture and cook until thickened, about 1 minute. Season the gravy with salt and pepper and keep warm over low heat.
Once your potatoes have cooked in the butter with the lid on for about 8 minutes, remove the lid and, shaking the pan occasionally, let them continue to cook and brown on all sides, about 5 more minutes.
While your potatoes are browning, salt the pan of boiling water and add the veggies. Cook until tender, about 2 minutes. Remove them from the pan and reserve until you're ready to serve.
Once the potatoes have browned, toss the chives into the pan and give it a shake to coat the potatoes. Serve each chicken breast with a side of potatoes and steamed veggies and a hearty ladle of the pan gravy over everything.