Begin by setting the oven to 375°F, positioning the rack in the center.
Prepare the Chicken: Season the chicken breasts thoroughly with 1 teaspoon of the sea salt. In a large, ovenproof skillet or braiser, heat 2 tablespoons of olive oil over medium-high heat until shimmering. Place the chicken in the skillet and sear until golden brown, about 3 minutes per side. Remove the chicken from the skillet and set it aside.
Mix the Honey-Mustard Sauce: In a small mixing bowl, combine the whole grain mustard, Dijon mustard, honey, remaining ½ teaspoon of salt, apple cider vinegar, paprika, and garlic powder. Stir until smooth and well-blended, then set the sauce aside.
Roast the Vegetables: Add the remaining 2 tablespoons of olive oil to the same skillet, heating until it shimmers. Add the chopped shallots, potatoes, and rosemary, tossing to coat them evenly in oil. Place the skillet, uncovered, into the oven and bake for about 15 minutes, or until the potatoes begin to tenderize.
Combine Ingredients: Take the skillet out of the oven and add the green beans to the softened potato mixture. Nestle the seared chicken breasts back into the skillet, ensuring they are evenly spaced. Pour the prepared honey-mustard sauce over the chicken, green beans, and potatoes, tossing gently to coat.
Bake Until Done: Return the skillet to the oven and bake for an additional 25-30 minutes, or until the green beans and potatoes are tender, and the internal temperature of the chicken reaches 165°F on an instant-read thermometer.
Slow Cooker Method
In a small bowl, mix the whole grain mustard, Dijon mustard, honey, ½ teaspoon of salt, apple cider vinegar, paprika, and garlic powder.
Season the chicken with the remaining teaspoon of salt. Add the chicken, shallots, potatoes, rosemary, green beans, and honey-mustard sauce to the slow cooker, tossing to combine. Cover and cook on LOW for 6-8 hours or on HIGH for 4-5 hours. If you prefer your green beans slightly firmer, add them to the slow cooker about 2 hours before the end of the cooking time.