In a large skillet with a lid, heat oil over medium heat. Add the sliced garlic, anchovies and red-pepper flakes and cook until the garlic turns golden brown at the edges, about 1 min. Stir in tomatoes, salt, pepper and basil sprig, then turn heat to medium-low.
Simmer until the tomatoes break down and thicken into a sauce, 20-25 mins. Stir in Parmesan, butter, salt and red pepper flakes to taste.
Using the back of a spoon, make 6 divots into the tomato sauce, then crack an egg into each divot. Cover the pan and let cook until the eggs are set, about 2 to 3 mins for runny yolks or longer based on your preference. (If the pan is not covered, the eggs won't cook through.)
Meanwhile, toast the bread then rub with the cut garlic clove, drizzle with oil, and sprinkle with salt.
To serve, sprinkle eggs with more Parmesan and chopped herbs, then spoon onto plates or into shallow bowls. Serve with garlic toast and pass pepper flakes at the table.