In a large bowl, combine the yeast and warm water. Stir to dissolve the yeast and allow the mixture to rest for 5 minutes.
Add about half of the flour over the yeast mixture and mix.
Add the salt, pepper and remaining flour and mix until incorporated.
Lightly flour a cutting board or flat surface. Knead for 3 to 5 minutes.
Let rest in an oiled bowl for about an hour.
Knead for a couple of minutes.
Wrap in plastic wrap and place in the fridge for 30 minutes.
Meanwhile, saute the onions and mushrooms in 1 tbsp of olive oil for 4 - 5 minutes. Season with salt and pepper. Set aside.
Roll out the dough to the size of your cast iron skillet.
Heat the skillet on medium high heat until smoking.
Add the dough and cook for a couple of minutes.
Drizzle the remaining olive oil on top of the dough.
Add the cheese, then the mushroom, onions, green peppers and cook for a few minutes.
Crack the eggs on top of the pizza. Place into a 425 degree oven.
When the eggs are set, remove from the oven.
Top with the arugula.
Serve and enjoy!