Bring the first 3 ingredients to a simmer in a medium saucepan (do not boil). Keep warm over low heat.
Heat oil in a large saucepan over medium-high heat. Add onion, and sauté 3 minutes. Add rice; sauté 2 minutes. Stir in Champagne; cook 1 minute or until the liquid is absorbed. Stir in 1 cup broth mixture; cook for 5 minutes or until liquid is nearly absorbed, stirring constantly. Reduce heat to medium. Add remaining broth mixture, 1/2 cup at a time, stirring constantly until each portion of broth is absorbed before adding the next (about 25 minutes total). Stir in feta. Sprinkle each serving with Parmesan cheese.