For the pastry: Place flour in a large bowl, make a well in the center, and add oil, salt, and 2 cups water. Mix thoroughly with a wooden spoon. When dough becomes too stiff to stir, knead on a floured work surface until smooth and soft, about 20 minutes (dough will be quite sticky). Divide into 6 equal balls, wrap in plastic wrap, and refrigerate for at least 2 hours.
For the filling: Bring a large pot of salted water to a boil over high heat. Add swiss chard and cook until tender, about 3–4 minutes. Drain in a colander and refresh under cold running water. Squeeze out excess water and finely chop.
Heat half of the oil in a large skillet over medium heat. Add swiss chard and cook, stirring, for 3 minutes. Transfer to a large bowl. Cool swiss chard slightly, and then add parmigiano-reggiano, ricotta, and oregano. Mix thoroughly, stir in cream. Season to taste with salt and pepper. Set aside.
Preheat oven to 400°. Lightly grease a 10" pie pan or 12" pizza pan. On a floured surface, roll out a ball of dough as thinly as possible (about 13 ½" in diameter) and transfer to pan. Brush with a small amount of oil, then repeat with 2 additional balls of dough, layering them on top of the first one.
Spread swiss chard mixture on top of rolled dough, allowing a 2" border around outer edge. Using the back of a spoon, make 5 indentations in the swiss chard mixture. Break an egg into the indentations, then top each with ½ tbsp. of butter. Season with salt and pepper.
Roll out remaining balls of dough as above, lay on top of filling, and brush each one with oil. Twist outer edge of dough together to seal. Bake torta until golden, about 45 minutes. Serve in wedges, warm or at room temperature.