Make dough: Put yeast, oil, and 2 cups plus 1 tbsp. cold tap water in bowl of a stand mixer and mix, using dough hook, on lowest speed, 5 minutes or until yeast has completely dissolved. Add flour and mix another 8 minutes.
Cover bowl loosely with a dampened towel and let dough rise 20 minutes.
Add salt and mix on low speed until incorporated and dissolved, about 7 minutes.
Turn dough onto a lightly floured work surface and cut into 6 equal portions. Roll each portion with a circular pressing motion into a tight ball. Place on a lightly floured baking sheet.
Cover tightly with plastic wrap and let rise at least 4 hours at warm room temperature. Dough balls have risen properly when they are soft, pillowy, and full of air.
Heat a pizza stone or baking sheet on lowest rack of oven at 550° (or as high as oven will go), at least 30 minutes.
Make toppings: Cut green onions into long slivers, soak in cold water for 20 minutes, and drain in a colander. Set aside.
Bring a small pot of water to a boil, add guanciale, and boil until softened, about 5 minutes. Drain and set aside.
Working with 1 dough ball at a time (keep the rest tightly covered), set on a well-floured pizza peel or baking sheet. To stretch into an 11- to 12-in. circle, first tap down the center of the ball with your fingertips to gently deflate it. Next, push it outward from the center with your fingertips. Then pick up the dough circle and, holding it under the rim, turn it like a steering wheel, letting the gravity of the dough help it stretch. Drape the dough over the backs of your hands and gently stretch outward, rotating periodically. Flop the stretched dough down onto the pizza peel.
Spread about 1/2 cup pecorino evenly over dough. Top with a third of green onions, then a third of guanciale. Grind on some pepper and drizzle with 3 tbsp. cream. Give the peel a good shake every few seconds to keep dough from sticking.
Plant tip of pizza peel (or long edge of baking sheet) on pizza stone and shove pizza quickly onto stone (or preheated sheet). Bake 2 minutes, then remove from oven with pizza peel or baking sheet, crack an egg onto center, gently push back into oven from peel with your fingers, and bake 3 minutes more. Just before serving, drag a knife through the egg to spread it around.
Repeat with 2 dough balls and remaining toppings (top remaining 3 dough balls differently or freeze).
Make ahead: Form dough into balls (through step 4) and chill overnight, tightly covered with plastic wrap, instead of letting rise on counter (it will rise slowly in the fridge). You can also freeze dough for up to 2 weeks (let chilled or frozen dough come to room temperature before stretching).
*Find at well-stocked grocery stores and Italian markets.
Note: Nutritional analysis is per slice.