Lightly toast then grill the bread in a large skillet with enough olive oil to coat both sides of the toast. Grill till sizzling and golden.
In the meantime, in a separate skillet, fry the eggs to your liking -- over easy or sunny side up. Season with sea salt and fresh ground pepper.
Spread a generous slather of pesto on each grilled toast. Top with an egg and sprinkle with Parmesan.