Peel, quarter, and cut turnip into ¼" slices. Trim ends off green beans.
Cook the Turnips
Place a medium non-stick pan over medium heat. Add butter and allow to melt. Continue cooking until butter begins to smell nutty and turns golden brown, 2-3 minutes. Butter browns shortly after it stops sizzling, so listen closely for when it's ready. Add turnips and a pinch of salt and stir occasionally until lightly browned, 3 minutes. Add milk, bring to a boil, and stir occasionally until tender, 4-5 minutes.
Sear the Chicken
Pat chicken breasts dry, and season both sides with mesquite seasoning. Place a medium pan over medium heat and add 2 tsp. olive oil. Add chicken to hot pan and cook undisturbed until lightly browned, 2-3 minutes per side. Transfer chicken to one third of prepared baking sheet. Reserve pan; no need to wipe clean. While chicken sears, scoop turnip mixture into prepared casserole dishes and top with Parmesan. Place casserole dishes on another third of baking sheet.
Bake Gratins and Chicken
Place green beans on empty third of baking sheet and toss with 1 tsp. olive oil and a pinch of salt and pepper. Massage oil into green beans, then spread into a single layer. Roast until cheese is browned, green beans are tender, and chicken reaches a minimum internal temperature of 165 degrees, 8-10 minutes. Rest chicken and gratin 5 minutes. While chicken and green beans roast, make sauce.
Make Sauce and Finish Dish
Return pan used to sear chicken to medium heat. Add ¼ cup water and demi-glace and stir. Bring to a boil and reduce by half, 1-3 minutes. Place chicken on a plate. Place gratin next to chicken. Place green beans next to chicken. Place sauce in front of chicken.