Melt butter in a large heavy saucepan over medium heat; add green onions and thyme, and sauté 1 minute or until onions are soft. Add rice, stirring to coat. Add 2 1/2 cups broth, and bring to a boil. Reduce heat to medium-low; simmer, stirring occasionally, 15 minutes. Add 1/2 cup broth, and cook, stirring constantly, until liquid is absorbed. Repeat procedure with remaining broth, 1/2 cup at a time. (Total cooking time is about 20 minutes.)
Stir in asparagus. Simmer 3 to 5 more minutes or until asparagus is tender and mixture is creamy. Stir in Parmesan cheese. Serve immediately. Garnish, if desired.