Preheat the oven to 350°F.
Place the cleaned mushroom caps on a rimmed baking sheet. Brush the mushroom caps with 1 tablespoon oil, then season with ½ teaspoon sea salt and ¼ teaspoon pepper, divided evenly amongst the mushrooms.
Add the 3 tablespoons ghee or olive oil to a large pan over medium heat, then add hash browns and season with remaining ½ teaspoon sea salt and 1/4 teaspoon pepper. Cook 5 to 6 minutes, then flip, and cook an additional 3-4 minutes, or until the potatoes are crispy and cooked through.
Fill the mushroom caps with the hash browns, then add the chorizo to a large pan over medium heat (you may use the same pan you cooked the hash browns in) and cook for 5-7 minutes until browned, breaking up the mixture into small crumbles as it cooks.
Divide the chorizo evenly between the mushroom caps, layering on top of the hash browns.
Make a small well in the chorizo and hash browns on each mushroom, then crack an egg. Season with salt and pepper to taste.
Bake at 350°F for 18 minutes for a runny egg yolk, and 25 minutes for a hard egg yolk.
While the mushroom caps are cooking, blend together the cilantro, lemon juice, olive oil, salt, and pepper until smooth.
Remove the mushroom caps from the oven, let cool, then drizzle with the cilantro sauce.