For the pizza dough: Combine the warm water and yeast in a small bowl and whisk to dissolve. Let sit until it starts to foam, 7 to 10 minutes.
Meanwhile, in a separate bowl, stir together the all-purpose and whole-wheat flours, olive oil and salt and mix well. Stir the yeast mixture into the flour and mix with a wooden spoon until just combined. (The mixture should pull into a smooth ball.) Grease a clean bowl with olive oil and transfer the dough to the bowl. Cover with plastic wrap or a clean kitchen towel and set aside on the countertop for 45 minutes.
Place a pizza stone or ovenproof nonstick pan in the oven and preheat the oven to 550 degrees F.
Transfer the dough to a floured work surface. Punch the dough down by pressing the air out of it with floured hands. Gently knead the dough and divide it into 2 pieces. Roll 1 piece of dough into a ball under the palm of your hand, then roll it into a flat disc with a rolling pin. Finish stretching the dough by hand. Repeat with the remaining ball of dough. The pizzas are now ready to top.
Spread tomato sauce evenly on the pizza crusts. Slice the mozzarella into thin pieces and distribute evenly on top of the sauce. Add the chicken, vegetables, prosciutto and cherry tomatoes and drizzle with oil and vinegar. Spoon on the Bacon and Sweet Onion Marmalade and sprinkle with Parmesan. Bake each pizza until bubbling and the crust is golden brown, 7 to 10 minutes. Let cool for 3 minutes, top with arugula and serve.
Bacon and Sweet Onion Marmalade:
Cut the bacon into pieces small pieces and add to a large saute pan. Cook over medium heat until the bacon is browned, but not completely crisp.
Add the onions, thyme and garlic to the pan and cook, stirring often, until very tender, 12 to 15 minutes. Drain any fat that remains in the pan and deglaze with the vinegar. Scrape all the fond (the browned bits) from the bottom of the pan and stir in the sugar. Continue cooking until the pan is dry. Set aside to cool. The marmalade may be stored, covered in the refrigerator, up to 3 weeks.