Combine 1/4 cup kosher salt and 4 cups warm water in a glass dish and stir until the salt dissolves; add the chicken and brine at room temperature for 15–30 minutes. Remove chicken, rinse under cold water, and pat dry.
Whisk together honey, Dijon mustard, stone-ground mustard, yellow mustard, and 1/3 cup honey (total already includes honey) until smooth; set aside 1/4 cup of the sauce to coat the chicken and reserve the rest for serving.
Place potatoes in a microwave-safe dish with about 1/4" water, cover, and microwave 6 minutes until starting to soften; drain and transfer potatoes to a lightly greased 9x13-inch baking dish.
Toss potatoes with 1 tablespoon olive oil, 3/4 teaspoon salt, and 1/4 teaspoon pepper. Push potatoes to the sides of the pan to make a space down the center for the green beans.
Toss green beans with 1 tablespoon olive oil, 1/4 teaspoon salt, and 1/8 teaspoon pepper, then arrange them down the center of the baking dish between the potatoes.
In a large bowl combine 1 tablespoon olive oil, garlic powder, onion powder, paprika, dried parsley, 1 teaspoon salt, 1/2 teaspoon pepper, and 1/8 teaspoon cayenne to form a wet rub. Add chicken and rub the spice mixture into both sides.
Coat both sides of each chicken piece with the reserved 1/4 cup honey mustard sauce, then place the chicken on top of the green beans in the baking dish. Spoon any remaining sauce and spices from the bowl over the chicken.
Cover the pan with foil and bake at 400°F for 25 minutes. Uncover and check doneness with an instant-read thermometer; continue baking uncovered as needed until the chicken reaches 165°F (additional 10–20 minutes depending on thickness).
Let the chicken rest 5–10 minutes before slicing and serve with the remaining honey mustard sauce.