In a medium saucepan with a good lid, melt 1 tablespoon of the butter over medium heat. Break the spaghetti up and add to the pan. Cook, stirring a bit, until browned; 2 to 3 minutes. Add the rice and 1 ¾ cups chicken stock and bring to a boil. Cover, lower the heat to a simmer, and let cook for 18 minutes or so, until most of the stock has been absorbed. Fluff the rice with a fork.
Meanwhile, steam the green beans and set aside. Stir the green beans into the cooked rice, leave covered and set aside.
While the rice is doing it's thing, trim the chicken if needed and season with salt and pepper.
In a large skillet over medium-high heat, add the olive oil and swirl to coat pan. Add the chicken, and cook until golden, about 6 minutes per side depending on thickness. Transfer the chicken and cover while you make the sauce.
Melt the remaining 2 tbs butter in the now empty skillet and add the shallots and mushrooms. Cook for about 5 minutes until everything is nice and tender. Season with salt and pepper and sprinkle the flour over top. Stir and cook for 1 minute. Whisk the remaining 1 cup of chicken stock (or add a splash of white wine) and deglaze the pan. Stir in the sour cream, tarragon, Dijon, and any juices the chicken may have accumulated. Whisk until smooth.
Serve the chicken with the rice pilaf and a allow dinner guests to spoon the sauce over top.