For the dough, in a large bowl, combine the water, yeast, honey and 2 tablespoons of the oil and let stand until it begins to bubble, about 5 minutes. (If it doesn’t bubble or foam after 10 minutes, it means the yeast is no longer active and you’ll need to start again with a fresh packet.)
Combine 3 cups of the flour with the salt then add it to the yeast mixture, stirring with a wooden spoon, until it is all incorporated. Continue adding the remaining 1 cup flour 1/4 cup at a time, working the dough after each addition, kneading it with your hands, until it has all been incorporated. The dough will be slightly sticky. Transfer the dough to a lightly floured work surface and knead several more times, until the dough comes together and is smooth.
Add the remaining 1 teaspoon oil to a clean large bowl. Place the dough in the bowl and turn to coat the oil on all sides. Cover the bowl with plastic wrap or a kitchen towel and set in a warm, draft-free place until the dough nearly doubles in size, about 2 hours.
Remove the dough from the bowl and knead it briefly on a lightly floured surface. Divide the dough into four pieces, shape each into a ball, cover with plastic wrap or a kitchen towel, and allow them to rest 15 minutes. Position the rac6 in the bottom third of the oven and line with bricks, pizza tiles or a baking stone. Preheat the oven to 500°F for at least 1 hour before baking the pizzas.
Lightly flour a work surface and stretch a round of dough out to a 9-inch circle by gently tapping and turning it, leaving a lip on the edge. Once the center is large enough to fit the palm of your hand, continue to shape the dough by pressing, turning and pulling it gently with your palm.
Lightly sprinkle a pizza paddle or rimless cookie sheet with cornmeal and transfer one of the dough rounds to the paddle. Brush the dough with 2 tablespoons of the oil and arrange 4 slices of the mortadella on the pizza. Top with one-fourth of the Taleggio (make sure to sprinkle it in an even layer-don’t pile it in the center), one-fourth of the chiles and about 2 tablespoons of the Parmigiano.
Slide the dough onto the hot bricks, tiles, or stone and bake until the crust is golden and the cheese is bubbly, about 10 minutes. Remove from the oven, cut into slices, garnish with additional Parmigiano if desired, and serve immediately. Repeat with the remaining ingredients.