In a medium saucepan, melt the butter (or heat the olive oil) over medium heat. Add the broken angel hair spaghetti and sauté until it turns golden brown, stirring occasionally.
Once the spaghetti is golden, add the uncooked brown rice to the saucepan. Stir well to combine the rice and pasta, allowing them to toast together for about 1-2 minutes.
Carefully pour in the chicken broth (or water and chicken bouillon). Give it a good stir to ensure everything is well combined. Bring the mixture to a boil.
Once boiling, reduce the heat to low, cover the saucepan, and let it simmer for about 10 minutes, or until the rice is tender and the liquid is absorbed.
After the cooking time, remove the saucepan from heat. Let it sit covered for an additional 5 minutes to steam. Then, remove the lid and fluff the rice pilaf with a fork. Garnish with chopped parsley if desired and serve warm.