In a 3-quart saucepan cook onion and garlic in hot oil over medium heat for 3 to 5 minutes or until onion is tender, stirring occasionally. Add the rice; cook for 3 to 5 minutes or until rice is golden brown, stirring frequently.
In a large skillet, heat 1 tablespoon olive oil. Add onions; cover and cook over medium-low heat for 15 minutes. Uncover; cook 10 to 15 minutes more or until onions are browned, stirring frequently.
Meanwhile, in a 1 1/2-quart saucepan bring broth to boiling; reduce heat. Cover and keep broth simmering. Carefully stir 1/2 cup of the broth into the rice mixture. Cook over medium heat until liquid is absorbed, stirring frequently. Continue adding broth, 1/2 cup at a time, stirring frequently until the broth has been absorbed.
With the last addition of broth, stir in onions and asparagus. Stir until rice reaches a tender and creamy consistency. (This should take 20 to 25 minutes total.)