Sauté first 3 ingredients in oil until tender; set aside.
Combine meat, cumin, and oregano in a large Dutch oven; cook over medium heat until meat is browned. Add onion mixture, chili powder, tomatoes, and water; bring to a boil. Reduce heat; cover and simmer 2 to 3 hours, stirring frequently.
Ladle into individual soup bowls. Top each serving with cheese.