Melt butter in olive oil over medium-high heat. Add peppers, onions, and salt, and cook, stirring occasionally, until vegetables begin to soften and start to brown around the edges.

Stir in oregano, chili flakes, black pepper, and canned tomatoes, and bring to a simmer. Reduce heat to low, cover, and cook until vegetables are tender.

Stir in half the cheese until it melts. Add the eggs, break the yolks with the tip of a spoon, and stir into the tomato mixture to achieve your desired texture.