Pre-heat the oven to 375 degrees.
Wash the chanterelles very well, they tend to be pretty dirty. Pat the mushrooms dry and slice. Heat 1½ teaspoons of olive oil in a medium pan. When hot add the chanterelles and minced garlic, and a pinch of kosher salt. Sauté for about 5 minutes until lightly brown. Remove from the heat and briefly set aside.
Heat a grill or grill pan over medium high heat. Brush both sides of each pita with olive oil and place them on the grill. Grill for about 3 minutes on each side until grill marks appear. (If you do not have a grill/grill pan, don’t panic. You can lightly brown the pitas in a cast iron skillet or whatever pan you happen to own, okay?)
While the pitas are grilling, tear the prosciutto into roughly 2-inch pieces.
Place the grilled pitas onto a baking sheet. Spread each pita with ¼ cup marinara sauce and sprinkle with ¼ cup shredded mozzarella cheese.Divide the chanterelles and prosciutto between the to pizzas. Transfer the pizzas to the oven and bake for 8-10 minutes until the cheese is melted and the prosciutto has crisped at the edges.
While the pizzas are in the oven, fry your eggs. Heat two teaspoons of olive oil in a non-stick pan over medium heat. When hot, carefully crack each egg into the pan. Let the eggs cook for 3-4 minutes until the white are set, but the yolk is still runny. (If you don’t like runny eggs, cook them longer. Duh.)
Remove your pizzas from the oven and top each with ½ cup baby arugula and a fried egg. Sprinkle with shaved Parmesan and crushed red pepper flakes, and serve immediately.