Preheat oven to 300 degrees F. In a 12-inch skillet heat 2 tablespoons of the oil over medium-high heat. Dip tortillas, one at a time, into the oil just until hot. Drain on paper towels (do not stack), reserving oil in skillet. Keep four of the tortillas warm on a baking sheet in the oven. Set aside the remaining tortilla.
For salsa, add onion and garlic to the skillet; cook over medium heat for 2 to 3 minutes or until tender. Stir in tomatoes, chile peppers, cilantro, and cumin. Bring to boiling; reduce heat. Simmer, uncovered, for 5 minutes. Transfer mixture to a blender or food processor. Tear the reserved tortilla into pieces; add to blender or food processor. Cover and blend or process until mixture forms a coarse puree. Keep warm.
In the same skillet heat the remaining 1 tablespoon oil over medium heat. Carefully break eggs into skillet. When whites are set and edges turn white, add the water. Cover skillet and cook eggs to desired doneness (allow 3 to 4 minutes for soft-set yolks or 4 to 5 minutes for firm-set yolks).
To serve, place a warm tortilla on each of four serving plates. Top each with two fried eggs. Spoon warm salsa over eggs; sprinkle with cheese. If desired, garnish with green onions.