Put stock and 3 cups water into a medium pot and bring to a boil over medium-high heat. Meanwhile, melt 6 tbsp. of the butter in another medium pot over medium heat. Add noodles and cook, stirring constantly with a wooden spoon, until deep golden brown, about 10 minutes. Remove pot from heat, add rice, and stir until well combined.
Return pot with rice and noodles to medium heat, add boiling stock and salt and pepper to taste (do not stir), cover pot, and cook, undisturbed, until almost all the liquid is absorbed, 20–25 minutes.
Tuck the remaining 1 tbsp. butter into center of rice and noodles, about halfway down, using a fork to gently cover butter with some of the rice. Cover pot, reduce heat to low, and continue cooking until liquid is completely absorbed and rice and noodles are very tender, 10-15 minutes more. Remove pot from heat and set aside, undisturbed for 10 minutes. Using a large spoon, fluff rice and noodles well before serving.