Make the Dough: In a small bowl, stir the yeast into the lukewarm water. Set aside until the yeast dissolves, about 5 minutes.
In another small bowl, combine the cold water, sugar, salt, oil and honey. Stir to dissolve the sugar and salt.
Place 5¼ cups of the flour in a large bowl. Add the cinnamon and nutmeg and stir to combine. Make a well in the center of the flour and stir in the yeast mixture along with the cold water mixture. Using a wooden spoon, mix the dough, incorporating as much of the flour as possible. Turn the dough out on a lightly floured work surface and knead until soft and elastic, 10-12 minutes. It will still be a little sticky but shouldn't stick to your hands. (Add only a minimum amount of flour to the work surface to keep the dough from sticking).
Cut the dough in half to form two even portions. With floured hands, pick up one portion of dough and pull the opposite edges together, wrapping them underneath toward the center to form a tight smooth ball. Pinch to seal. Repeat with the second portion. Place each portion in a 1-gallon lock-top plastic bag. Squeeze out all the air and seal the bags, allowing enough room for the dough to double in size. Refrigerate for at least 10 hours or up to 2 days. Remove from the refrigerator 1 hour before using to allow to come to room temperature.
You will use one portion of the dough for this recipe.
Make the Pizza: Position an oven rack on the second-lowest level in the oven and place a baking stone on the rack. Position another rack in the upper third of the oven. Preheat the oven to 500 degrees.
While the oven is heating, make the apple topping: Place the nuts in a singe layer on a rimmed baking sheet. Bake the nuts until lightly browned, 5 to 8 minutes, depending on how hot the oven is. (Watch the nuts carefully; they burn quickly.) Set aside to cool.
In a large sauté pan over medium-high heat, melt the butter and swirl to coat the pan. Add the apples and sauté, stirring constantly, until well coated with butter but still crisp, 3 minutes. Add the brown sugar, cinnamon, nutmeg, salt, and raisins. Sauté, stirring frequently, until the apples are tender but still hold their shape and the liquid is reduced, about 5 minutes. Remove from the heat and set aside.
Coat a 14-inch pizza screen or perforated pizza pan with cooking spray. Remove the dough from the plastic bag and place on a lightly floured work surface. Lightly dust the dough with flour. Using a rolling pin, roll the dough into a 10-inch round without rolling over the edges. Shake the excess flour from the dough. Stretch the dough to a 14-inch circle and place it on the prepared pizza pan. Gently re-stretch the dough into a 14-inch round.
Scatter the mozzarella over the dough, leaving a 1-inch border.
Place the pizza in the oven on the upper rack. Bake the pizza until the crust is crisp and golden brown and the cheese is golden, 9 minutes. Transfer the pizza from the pan and place the crust directly on the baking stone. Remove the screen or pan from the oven. Bake the pizza until the bottom of the crust is golden brown, 2 minutes longer. Remove the pizza from the oven and transfer to a wire rack to cool slightly.
Stir the toasted walnuts into the apple mixture. Spread the apple mixture in an even layer over the pizza crust, leaving a 1-inch border. Slide the pizza crust directly onto the baking stone and bake the pizza until the apple mixture is heated through, about 5 minutes.
Transfer the pizza to a cutting board. Dust the pizza with powdered sugar.
Slice into wedges and serve with scoops of vanilla ice cream.