Cook's Note: You will need extra flour for rolling the dough.
For the dough: In a bowl, combine the yeast and warm water. Stir to dissolve the yeast and allow the mixture to rest for 5 minutes. Using a sieve or strainer, "sift" about half of the flour over the yeast mixture and blend until smooth with your hands. Add the salt, pepper and honey and mix to blend. Sift in the remaining flour and mix to blend. Lightly flour a cutting board or flat surface. Turn the pizza dough onto the floured area and knead for 3 to 5 minutes. The flour should feel smooth and the ingredients fully integrated. Add the dough to a lightly oiled bowl. Cover with plastic wrap and let it "rest" in a warm place until it has doubled in volume, about 1 1/2 hours. Press gently on the dough and turn it onto a floured surface. Divide the dough into 4 equal parts, rolling each quarter into a loose ball. Cover with a clean kitchen towel and allow the dough to "rest" for an additional 15 minutes. Flatten each ball, 1 at a time, and roll into a 6-inch round. Make the pesto: In a medium bowl, add the broccoli rabe, 2 tablespoons of the olive oil and salt and the paprika to taste; stir to combine. Heat a large saute pan over high heat and add the broccoli. Keep the heat high, allowing it to brown and char on the edges. When the broccoli rabe is tender when pierced with the tip of a knife, remove from the heat and allow it to cool. Put the broccoli rabe on a cutting board and finely chop. Transfer to a bowl. Put the garlic through a garlic press and add to the bowl. Stir in the Parmesan, 2 more tablespoons of the olive oil and the basil leaves. Taste for seasoning. Cook the mushrooms: In another saute pan, over medium heat, add the mushrooms and a tablespoon of the olive oil. Season with salt and garlic powder, to taste. Cook until they release their water, 8 to 10 minutes.
Preheat the oven to 400 degrees F. In a cast iron skillet, large enough to hold a pizza round, heat 2 tablespoons of the olive oil over medium heat. When the oil begins to smoke lightly, shut the heat off under the skillet and add 1 of the pizza rounds. Use a spatula to press gently on the top of the round to keep it relatively flat as it cooks. When you see it is slightly brown on the underside, season the top lightly with salt and pepper and flip the round to its second side. Season with salt and pepper. Repeat with remaining dough. Finish the pizzas: Arrange all of the browned rounds on baking sheets and layer with the mozzarella slices, broccoli rabe pesto and a sprinkling of mushrooms. Bake in the oven for a few minutes until the cheese melts. Season with salt and serve.