Warm Chicory with Eggs and Pecorino Romano

Ingredients

1 medium-size red onion, halved and sliced
1/4 teaspoon kosher or sea salt
2 tablespoons olive oil
1 pound chicory, rinsed and coarsely chopped
4 garlic cloves, minced
1/4 teaspoon red-pepper flakes
4 large eggs
1/4 cup (1 ounce) shredded Pecorino Romano cheese
Freshly ground black pepper