Combine first 3 ingredients in a large saucepan. Bring to a simmer; slowly whisk in polenta. Bring to a boil. Cover, reduce heat, and simmer 5 minutes, stirring occasionally. Remove from heat.
Melt the butter in a large nonstick skillet coated with cooking spray over medium-high heat. Add mushrooms; sauté 5 minutes or until lightly browned. Add oil, asparagus, and ham to pan; sauté 4 minutes or until the asparagus is crisp-tender.
Serve polenta topped with asparagus mixture; sprinkle with cheese.