Place a pizza stone on the middle rack of the oven and preheat the oven to 500 degrees F.
On a lightly floured surface, stretch or roll out each of the balls of dough into an 8-inch circle. Brush each of the circles with the chili oil and start to layer, dividing the ingredients among the 4 pizzas. Start with the tomatoes, then the mozzarella and Fontina cheeses, the herbs, chicken, and Parmesan. Bake until the pizza crusts are nicely browned, 10 to 12 minutes.
Toss the lettuce with the vinaigrette and season lightly with salt and pepper. Remove the pizzas from the oven, arrange the lettuce on each of the pizzas, and top with some of the shaved Parmesan. Cut into slices and serve immediately.
Dissolve the yeast in the warm water. Add the honey and let sit for 5 minutes, until foamy. Put the flour in a food processor. Mix the 1 cup cool water with the olive oil and salt. With the motor running, pour the olive oil mixture and the yeast slowly in through the feed tube. Process until the dough forms a ball on the blade. Transfer the dough to an oiled bowl, cover and let rise until double in bulk. Punch down the dough and knead it on a lightly floured surface for 1 minute. Divide the dough into 4 equal portions and roll them into tight balls. Place on a tray, cover with a damp towel and let rest for several hours or overnight in the refrigerator.
Roll or stretch each bowl of dough into a 7 to 8-inch circle. Place the circles, 1 at a time, on a wooden peel or on a baking sheet and build the pizza as desired.
Preheat the oven to 250 degrees F.
In a pot of boiling water, blanch tomatoes. Drain and refresh in ice water. Drain. Peel, core, cut into quarters, and remove seeds.
Line a baking tray with parchment paper and arrange the tomato quarters on the tray, cut side down. Drizzle generously with olive oil. Sprinkle with thyme and garlic. In a small bowl, combine salt, pepper and sugar, and sprinkle evenly over the tomatoes.
Bake until the tomatoes begin to shrivel, about 1 hour. When the tomatoes are cool enough to handle, transfer to a container. Pour olive oil over and cover the container. Refrigerate and use as needed.
Arrange the cubed chicken in a shallow medium bowl and toss with 1/2 cup olive oil, lime juice, jalapenos, garlic, and cilantro. Season with salt and let marinate for 1 hour, refrigerated.
In a skillet or saute pan large enough to hold the chicken in 1 layer, heat the remaining tablespoon of oil. Remove the chicken from the marinade with a slotted spoon. Saute the chicken just to brown on all sides. Do not overcook. Remove from the skillet with a slotted spoon and set aside.
In a medium bowl, whisk together the egg, lemon juice, garlic, Worcestershire sauce, red pepper flakes, mustard, and anchovies. Slowly whisk in the oils to emulsify. Stir in the cheese and season with salt and pepper. Refrigerate, in a covered container. When ready to use, whisk again.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.