Heat the oil and butter in a large, heavy pot over low-medium heat. Add the onions, tomato and its juices, jalapeño pepper, salt, and pepper. Cover and cook for about 10 minutes, stirring occasionally until the onions and tomatoes have softened and released some of their liquids.
Add the dandelion greens and sprinkle with additional salt. Using tongs, turn the greens to coat with the oil. Cover and cook for 10 minutes, stirring occasionally, until the greens are wilted and have released some of their liquids.
Stir in 8cups of water, cover the pot, and raise the heat to high. Bring to a boil. (This will take about 5 minutes.) Reduce the heat to medium. Cover and cook at a low-medium boil, stirring occasionally, for about 20 minutes.
Add the broken spaghetti, cover, and cook for about 7 minutes, stirring occasionally until the spaghetti is barely tender.
Stir in the beaten eggs and cook, uncovered, stirring frequently for 1-2 minutes, until the eggs are just set. Taste for seasonings. Serve hot, with grated cheese on top, if desired.