In a large cast iron pan or saucepan, heat the olive oil over medium heat. Add the anchovy paste (or finely chopped anchovy fillets) and crushed red pepper, stirring until the anchovy dissolves into the oil and becomes fragrant, about 1 to 2 minutes. Stir in the garlic and cook just until fragrant, then add the tomato paste and continue cooking, stirring, until it deepens in color, another 2-3 minutes.
Pour in the crushed tomatoes with their juices and stir in the olives, capers and thyme or basil sprigs. Season lightly with salt and pepper and bring the sauce to a gentle simmer. Reduce the heat to low and cook, stirring occasionally, until thickened and slightly reduced, about 20 to 30 minutes. Taste and adjust the seasoning as needed, then discard the herb sprigs.
Using the back of a spoon, make small wells in the sauce and crack the eggs into them one at a time. Cover and cook until the whites are set and the yolks reach your desired consistency. Finish with a drizzle of extra-virgin olive oil and, if you like, a small splash of vinegar before serving.