In a 4 qt. saucepan over low heat, melt the butter. Add the onions, garlic, oregano, and bay leaves and cook, covered, stirring once or twice, for 10 minutes. Stir in the chicken broth, potatoes, salt and pepper and bring to a boil. Lower heat and simmer, partially covered, stirring twice, until the potatoes are very tender, about 25 minutes.
Cool soup slightly, discard bay leaves, and puree in food processor. Return soup to pan, stir in green chiles, and set over medium heat. Bring to a simmer and cook, uncovered, stirring frequently, until the soup has thickened, about 15 minutes. Stir in the cream and adjust the seasoning. Can be prepared up to 3 days ahead. Rewarm over low heat, stirring often.
Ladle soup into wide bowls, sprinkle the cheese over the soup, and scatter the tortilla strips over the cheese. Serve immediately.