This is an all-time favorite breakfast/brunch dish. It's so good that there really aren't words to describe it and if you tried to use any that you know in an effort to do this dish justice, you'd probably regret it.
Italian or French bread
2-4 oz. Parmesan cheese, divided (do I even need to say it - nothing from a can!)
1-2 Tbsp. butter
For the pesto
2 Tbsp. slivered almonds, toasted
Fresh Italian parsely - 1 large fistful
Fresh basil - 1/2 to 2/3 the amount of parsely used
4 -5 cloves garlic
1/4 c. extra virgin olive oil
1/2 tsp. Kosher salt
Fresh ground black pepper
As you've noticed, the amount of eggs and bread has not been specified. The amount is a function of how much you want to make, how many you are feeding, etc.
Preheat oven to 400 degrees Fahrenheit. Slice bread (bias cut) into desired number of 1/4 to 1/2 inch thick slices and brush lightly with olive oil or real butter (not that the two are interchangeable, it just depends on the effect you prefer) on one side. Toast in oven til light to golden brown, remove and set aside. Toast slivered almonds in oven on a small baking dish for 3 minutes, being careful not to let them get too over-browned or burnt.
Wash parsley and basil well under cold water and pat dry with paper towels. Place herbs in blender with toasted almonds, garlic, olive oil and 1/4 of your grated Parmesan. Pulse blender several times until herbs, garlic and nuts are well pulverized but still a bit coarse (not pureed). If you have to scrape down the sides of the blender between pulses, make sure blade has completely stopped spinning first.
Heat butter in non-stick skillet over medium-low heat. Fry desired number of eggs over-easy two-three at a time. Apply pinch of table salt and pepper to each egg. Apply pesto (about 1 Tbsp.) to each piece of toasted bread. Plate bread with pesto and top each piece of bread with an egg. Sprinkle each eggs with remaining Parmesan and another Tbsp. or so of pesto. Serve while eggs are warm.
Food for Thought
You may also do this with poached eggs. No instructions for poaching eggs will be given here but it is easy to find in various print and web resources. Those who know how to poach eggs well may agree that the process is something akin to making a potion in Harry Potter's advanced potions class where you require a personalized copy of the instructional text from the Half-Blood Prince himself, complete with helpful hints in the margins. Over-easy works great and is self-explanatory. The point is, that you have firm egg white with an intact yolk that is not so firm (which is not to say that the yolk is infirm).
My wife does not care for uncooked egg yolks (or so she thinks), yet she loves pasta carbonara - which she know employs uncooked egg yolks. Whatever. The pesto is, of course, herbacious as all get out but the addition of the almonds and Parmesan temper what might otherwise be overly grassy notes. You may wish to add a little additional salt to your finished product but try a bite first because the Paremesan in the pesto and on the eggs adds more saltiness than you might expect.
If you are concerned about foodborne illness from an uncooked egg yolk, that's OK. If you are elderly or pregnant, or have some other potential contraindication, you may want to avoid such a preparation. For the rest of us, it is relatively safe (I've been eating eggs over-easy since I was a kid and I'll let you know the first time I become ill from it). In fact, you are more likely to get salmonellosis or other foodborne illness from other foods which are generally consumed raw than from an uncooked egg yolk; certain vegatable matter like pre-packaged spinach, salad greens, sprouts, etc. In many countries, eggs may move from the chicken coupe to your home and never see the inside of a refrigerator until you get them home. And besides, it is delicious and adds a unique and wonderful element of rich and wholesome yumminess. Buon appetito!