1. Preheat the oven to 400. With a pastry brush, spread 4 teaspoons of the softened butter evenly over the bottom of four heatproof serving plates or individual shallow baking dishes about 6 inches in diameter.
2. Place a tortilla on each plate or dish and brush the tops with the remaining 4 teaspoons of softened butter. Set aside.
3. With a mortar and pestle, or in a small bowl with the back of a spoon, rub the chilies, garlic and salt to a smooth paste. Drain the tomatoes in a sieve set over a bowl and chop them coarsely.
4. Combine the chili-and-garlic paste, the tomatoes and their liquid, the oregano and the cumin in a heavy 10- to 12-inch skillet. Stirring constantly, bring to a boil over high heat.
5. Then cook briskly, stirring from time to time, until the sauce is thick enough to hold its shape almost solidly in a spoon. Remove from the heat and taste the sauce for seasoning.
6. Arrange the sliced artichoke hearts in a ring around the outside edge of each tortilla. Pour the tomato sauce over the tortillas, masking the artichoke-heartrings completely. Break 2 eggs into the center of each ring and bake in the middle of the oven for 2 or 3 minutes.
7. Then sprinkle each pair of eggs with ¼ cup of the cheese,and continue baking for 5 or 6 minutes longer, until the cheese has melted and the eggs are set.
8. Serve the Huevos Monterey at once, directly from the baking dishes. Mound the chopped onions in a small bowl and present them separately as an accompaniment.
