In a medium saucepan with a tight-fitting lid, melt 1 tablespoon butter over medium heat. Add the spaghetti and cook until browned, 2 to 3 minutes. Add the rice, then stir in 1 3/4 cups chicken stock and bring to a boil. Lower the heat, cover the pot and cook for 18 minutes. Stir in the green beans, cover and cook for 5 minutes. Remove from the heat and let stand for a couple of minutes, then fluff with a fork and season with salt and pepper.
Meanwhile, season the chicken with salt and pepper. In a large skillet, heat the EVOO, 1 turn of the pan, over medium-high heat. Add the chicken and cook, turning once, until golden, about 12 minutes. Transfer the chicken to a plate and cover with foil to keep warm.
Add the remaining 2 tablespoons butter to the skillet and lower the heat to medium. Stir in the mushrooms and shallot and cook until tender, about 5 minutes. Season with salt and pepper, sprinkle in the flour and cook, stirring, for 1 minute. Whisk in the remaining 1 cup chicken stock, the creme fraiche, tarragon, mustard and any chicken juices.
Slice the chicken and serve alongside or on top of the rice. Spoon the sauce on top.