Combine water, yeast, sugar, flour, oil, and salt in a bowl until a rough dough forms. Knead in a stand mixer with a dough hook for 6 minutes on medium speed. Divide into 3 balls (~300 grams each) and place in lightly oiled containers. Let rise for 3 hours (1.5 hours in the fridge, 1.5 hours at room temperature).
Preheat a wood-fired oven to 550°F/285°C or a home oven to its highest setting, 480–540°F/250–280°C, using an inverted baking sheet.
Lightly flour a work surface and stretch dough from the center outward, rotating gently until thin and round with slightly thicker edges.
Spread a thin layer of cream over each base. Sprinkle with black pepper, grate Parmesan, add mozzarella, and top with Portobello slices. Bake 10 minutes in a wood-fired oven or 12 minutes in a conventional oven until golden on top and bottom.
Remove, grate extra Parmesan if desired, slice, and serve immediately.