Heat the olive oil in a large skillet over medium-high heat. Add the garlic and cook until golden brown, about 3 minutes. Remove the garlic with a slotted spoon and discard. Increase the heat to high and add the salami, onion, peppers and red pepper flakes. Cook, stirring, until the salami starts to brown, about 5 minutes. Add the tomato puree and 1/2 cup water and cook until the vegetables are tender, about 5 minutes.
Reduce the heat to medium. Scatter the parsley leaves into the skillet. Make 4 indentations in the pepper mixture for the eggs. Break 2 eggs into a small bowl, then pour both into one of the indentations. Repeat with the remaining eggs. Sprinkle with the grated cheese. Cover and cook until the whites are set and the yolks are cooked to desired doneness. Serve with the bread.