Using a teaspoon, (I used a grapefruit spoon) gently scrape the black gills off the mushroom caps. Set a cooling rack over a rimmed baking sheet and arrange the scraped-out mushroom caps, scraped side down; bake for 5 to 7 minutes. The caps will release some water. Transfer mushroom caps to an ovenproof dish, scraped side up. This step prevents the mushrooms to releasing their liquid when baking.
In a small microwave safe bowl (or pan), add butter, minced garlic and parsley. Melt in microwave (or pan) for about 30 to 45 seconds on high allowing to infuse until garlic is fragrant. Allow to cool a little bit then brush the mixture on the outside and inside of the mushroom caps
In a bowl, combine cream cheese, salt, pepper, and grated cheese and cream together. Spoon the cheese mixture into each portobello caps smooth right to the edges, form a small hole in the center for the egg yolk.
One at a time, separate the eggs, reserving the whites for another use, and gently pour the yolk into each hole...(I also used a little white too).
Top each mushroom and egg with Parmesan cheese, and a pinch of black pepper. Pop back into the oven and bake for 4 to 5 minutes or until the cheese is melted, yolks should still be runny. Garnish with chopped parsley. Enjoy!