At least 3 hours ahead, start the pizza dough. In a small bowl, combine the water, yeast, and olive oil. In a large bowl, whisk together the flour, yeast, and salt. Add the wet ingredients to the dry and knead the dough until the mixture comes together into a ball, a bit tacky to the touch. Add more water as needed, or more flour if it’s too wet. Transfer the dough to a floured surface and knead to create a smooth ball of dough.
Transfer to an oiled bowl, turn dough once to coat in oil, and cover bowl with a clean, damp dish towel. Place in a warm location, or, if your kitchen is cold, inside your oven. Let rest until it doubles in size, about 2 hours. Once the dough is doubled in size, roll it into a ball. (Cut the dough in half and form into 2 balls if making 2 pizzas.) Will keep in the fridge for 2 weeks, or in the freezer for up to 3 weeks. Bring to room temperature before using.
Preheat oven to 525º Line a baking sheet with parchment paper and sprinkle with cornmeal. Spoon the tomato sauce over the pizza dough, leaving a ½-inch border, then distribute the mozzarella evenly. In quadrants, distribute 3 olives and 4 artichoke hearts in quadrant 1, the mushrooms in quadrant 2, and another 3 olives and 4 artichoke hearts in quadrant Leave quadrant 4 empty.
Bake for 8-10 minutes, until crust is beginning to brown. Remove from oven and crack egg in the fourth quadrant (angle egg towards center of pizza so it doesn’t run off, and be careful not to tilt your baking sheet in transfer). Return to oven and bake another 1-2 minutes, until whites are cooked but yolk is still runny. Remove from the oven when the edges are golden brown and crisp to your liking. Transfer to a wooden cutting board, sprinkle with red pepper flakes if desired, and eat immediately.