Bring a wide pot of water to a boil over high heat. Heat the broth in a large saucepan over medium heat.
Trim the asparagus by breaking off the tough ends (where they break naturally). Finely chop the onion to yield about 1 cup. Add the asparagus to the water and cook for about 3 minutes, until barely tender. Drain and cut crosswise into 2-inch pieces.
Heat 4 tablespoons of the butter in a large saute pan over medium heat. It will bubble and turn golden brown, smelling nutty. Keep the butter moving in the pan so its solids don't burn. Add the onion and stir to coat; cook for 4 or 5 minutes until it is softened and on its way to translucent.
Stir in the rice, then carefully pour in the wine, using a wooden spoon to scrape any browned bits from the bottom of the pan. Ladle 1/2 cup-increments of the hot broth into the pan, cooking and stirring each time until it is absorbed before adding the next. Use all the broth. The rice should be tender and creamy. Taste and season with salt if needed (keeping in mind that cheese will be added). Remove from the heat.
Stir in the cheese and the remaining tablespoon of butter, until the latter is completely incorporated. Gently stir in the asparagus pieces, then season with pepper to taste.
Divide among wide, shallow bowls. Serve warm.