In 2-quart saucepan, heat broth over medium heat until hot.
In 12-inch skillet, heat oil over medium heat. Cook onion and carrot in oil 1 to 2 minutes, stirring occasionally, until crisp-tender. Stir in barley. Cook and stir 1 minute.
Pour 1 cup of the hot broth over barley mixture. Cook uncovered about 5 minutes, stirring occasionally, until liquid is absorbed. Stir in asparagus. Continue cooking 15 to 20 minutes, adding broth 1 cup at a time and stirring frequently, until barley is tender and liquid is absorbed; remove from heat. Stir in remaining ingredients.