Walnuts were once thought of as “brain food,” but we already know you're wise: You're going to choose this chicken dish with its delectable, rich walnut sauce. The nutty, creamy flavor clings perfectly to every inch of this chicken. (Feel free to use the sauce on the green beans and potatoes, too; we won't tell such a smart guy or gal how to live their life.) Plus, this recipe is so easy to make, you'll barely need to use that big brain of yours.
Halve potatoes lengthwise. Trim ends off green beans. Peel and halve shallot. Slice one half into thin rounds and mince the other half. Finely chop walnuts. Pat chicken breasts dry, and season both sides with ¼ tsp. salt and a pinch of pepper.
Roast the Potatoes
Place potatoes on prepared baking sheet and toss with 2 tsp. olive oil and ¼ tsp. salt. Spread into a single layer and roast 10 minutes. Remove from oven. Potatoes will finish cooking in a later step. While potatoes roast, cook chicken.
Cook the Chicken
Place a medium non-stick pan over medium heat and add 1 tsp. olive oil. Add chicken to hot pan and cook until chicken reaches a minimum internal temperature of 165 degrees, 5-8 minutes per side. Remove chicken to a plate and tent with foil. Reserve pan; no need to wipe clean.
Roast the Beans
Add green beans and shallot rings to baking sheet and carefully toss with potatoes. Spread into a single layer and roast until shallot rings are aromatic and potatoes are browned, 10-15 minutes. While vegetables roast, make sauce.
Make the Sauce
Return pan used to cook chicken to medium heat. Add minced shallot and walnuts to hot pan and cook until aromatic, 1-2 minutes. Add cream and demi-glace, and cook until thick enough to coat the back of a spoon, 1-2 minutes. Remove from burner.
Finish the Dish
Place vegetables on a plate. Serve sauce on plate and top sauce with chicken.