Heat 1 tablespoon of olive oil in a medium (10-inch) sauté pan. Add the onions and cook over medium to medium-low heat for 5 to 7 minutes, stirring occasionally, until tender and starting to brown. Add the garlic and cook for one minute. Add the arrabbiata sauce, red pepper flakes, and rosemary, bring to a simmer, and cook over medium-low heat for 5 minutes. Remove and discard the sprig of rosemary.
Carefully crack one of the eggs into a small (4-inch) bowl and gently slide it into one corner of the pan (don’t break the yolk), using the edge of the bowl to make a slight indentation in the sauce as you pour the egg in. Repeat with the remaining 3 eggs, placing them on opposite sides of the pan. Cover the pan tightly and cook over medium-low heat for 4 to 6 minutes, until the egg whites are set but the yolks are still runny. Sprinkle with the Pecorino, parsley, sea salt, and black pepper, cover the pan again, and cook for one more minute.
To serve, use a large spoon to scoop up two eggs per person along with some of the sauce and carefully transfer to shallow bowls. Spoon the remaining sauce around the eggs and serve hot with the toasted bread.
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