Cut the tortillas into 1/4 inch-wide strips and leave them uncovered at room temperature until they are dry and crisp, about 24 hours (OR: Heat in iron skillet until dry and hot).

In a 4-quart 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, green chiles and potatoes. Season with salt, cumin and black pepper, and bring to a boil. Reduce heat and simmer, partially covered, stirring occasionally, for about 25 minutes (or until the potatoes are very tender).

Stir in the cream and adjust the seasoning if necessary. (The soup can be prepared up to 3 days ahead. Cool it completely and refrigerate, covered. Reheat it over low heat, stirring often, until steaming.)

Ladle the soup into wide bowls, sprinkle the cheese over the soup, and scatter the tortilla strips over the cheese. Serve immediately.