For the dough: Combine the flours, salt, sugar and yeast in a medium bowl. Mix in the water until combined. Cover with plastic wrap and let rise for 1 1/2 hours or up to overnight.
Position a pizza stone or inverted baking sheet in the oven and preheat to 475 degrees F.
Cut out 2 big sheets of parchment paper, and dust liberally with flour. Divide the dough into two pieces, and place each on a piece of parchment. The dough will be very sticky, so don't be shy in dusting it with enough flour as needed to handle it. Roll out on the parchment until the dough is very thin and about 12 inches in diameter, dusting with flour as needed.
For the toppings: Brush each round with 1 tablespoon olive oil and top them with the garlic, shallot, mozzarella, spinach and ham. Toss the potatoes with the remaining tablespoon olive oil in a medium bowl and distribute the potatoes over the pizzas in a single layer, not allowing the potatoes to overlap. Sprinkle everything with a good pinch of salt and a few turns of pepper.
Using a pizza peel or baking sheet, slide one of the pizzas on its parchment onto the pizza stone and bake until the cheese is splotchy with brown marks and the potatoes are browned, about 12 minutes.
Sprinkle with crushed red pepper, Parmesan and parsley. Repeat with the remaining pizza, and enjoy!