Place a medium nonstick skillet over low heat. Add the basil pesto and warm it gently until it begins to shimmer and lightly bubble.
Crack the eggs directly into the warm pesto, leaving a little space between them.
Cover the skillet and cook over low heat for about 2 minutes, or until the egg whites are set while the yolks remain soft and runny. For firmer yolks, cook for another minute or two.
Season the eggs lightly with salt, black pepper, and red pepper flakes or Aleppo pepper.
If using toast, spoon a small amount of warm pesto over each slice. Arrange the tomato and fresh mozzarella slices on top.
Carefully slide the eggs onto the prepared toast and spoon the remaining pesto over the top.
Serve this Pesto Eggs Recipe immediately while the eggs are warm and the mozzarella is creamy.