In a skillet over medium-high heat (or in the wood oven), sizzle about 2 tablespoons of olive oil until hot. Add the shallots and mushrooms and stir frequently until the ‘shrooms are wilted and all the excess liquid has cooked off. Stir in the spinach, season generously with salt and pepper; cover and let cook for 1 to 2 minutes more, until the spinach wilts. Remove from the heat and set aside for up to an hour, if desired.
Flour a board and your hands with rice flour, then dimple the central two-thirds of the dough ball with your fingertips to flatten slightly; never pat or press down on the rim (corniche). Stretch and pull the dough into a roughly 13-inch round or oval with a raised corniche (I like to make an hourglass shape for this one). Transfer the dough circle to a long-handled wooden pizza peel liberally sprinkled with fine cornmeal. Break each egg into a ramekin or saucer.
Working quickly, brush the shaped dough with a little olive oil and season lightly with salt and pepper. Gently spread the spinach-mushroom mixture over the dough in two circles, creating two empty depressions (where the eggs will go, and be held in by the low “walls” of the spinach mixture) and leaving a half-inch border. Scatter the shaved cheese over the spinach mixture and inside the egg-depressions. Jiggle the dough on the peel to make sure it’s still mobile, then quickly and confidently slide the pie onto the floor in the center of the oven and bake, rotating occasionally after the first minute or two.
Bake until the cheese has begun to soften and the crust is only just barely pale golden. Pull the pie toward you with a peel or a hook, and slide an egg from each saucer into each of the depressions you have created. Slide the pie back into the oven and continue cooking until the white of the egg is almost set but the yolk is still nice and runny, another 1 to 3 minutes, depending on the temperature. Retrieve with a metal peel and transfer to a cutting board. Season with finishing salt and let stand for 1 minute if desired. Slice and enjoy.