In a large saucepan cook barley in hot olive oil over medium heat about 5 minutes or until lightly toasted, stirring often. Add onion, butter, and garlic; cook until onion is tender.
In a small saucepan heat broth. Slowly add 3/4 cup of the broth to barley mixture, stirring constantly. Continue to cook and stir over medium heat until liquid is absorbed. Add another 1/2 cup of the broth, stirring constantly. Continue to cook and stir until the liquid is absorbed. Add another 1/2 cup of the broth; cook and stir until liquid is almost all absorbed.
Add asparagus and carrot; continue cooking until all the liquid is absorbed and the barley is tender. (Should take about 30 minutes total.)
Remove from heat. Stir in cheese and basil; season to taste with salt and ground black pepper. Makes 3 main-dish servings or 6 side-dish servings.