Place broth in small pot with heat on low. In a second pot, melt butter over medium heat. Add onion and cook, stirring, until soft (about 5 minutes). Add rice and stir until opaque (about 2 more minutes).
Add 1/2 cup portions of broth to rice stirring frequently, until liquid is absorbed. Repeat until there is no more broth. This should take about 20 to 25 minutes. Remove from heat and stir in cheese.