In a large bowl, combine yeast, 1/4 cup warm (110°) water, and 1/4 cup flour and stir to mix. Let mixture sit at room temperature until bubbles are visible, about 30 minutes.
Add remaining 1 3/4 cups flour, 1/2 cup plus 2 tbsp. warm water, the oil, and salt and, using a wooden spoon, stir until dough comes together in a rough mass.
Lightly flour a work surface and turn dough out onto it. Knead until smooth, elastic, and a bit tacky to the touch, 7 to 8 minutes. Shape into a ball. Oil a large bowl, place dough in bowl, and turn to coat on all sides with oil. Cover with plastic wrap and let dough rise in a warm place until doubled, 1 to 1 1/2 hours. (Or, let dough rise in the refrigerator overnight. The next day, bring it to room temperature before continuing.)
Make pizza: About 30 minutes before baking, place a pizza stone on lowest rack in oven. Preheat oven to 500° or to the highest temperature possible.
In a small bowl, combine garlic and 2 tbsp. oil and let stand 30 minutes.
In another bowl, whisk together remaining 2 tbsp. oil and the lemon juice and season with salt and pepper. In a medium bowl, combine fontina and mozzarella and toss lightly to mix well.
Lightly flour a work surface. Punch down dough. Divide it in half and form each half into a ball, but do not work the dough at all. Roll out 1 dough ball into a 10- to 11-in. circle about 1/4 in. thick. (Alternatively, press and stretch dough into a 10- to 11-in. circle.)
Lightly flour a pizza peel (or a rimless cookie sheet) and transfer dough circle to peel. Brush dough circle to within 1/2 in. of edge with half of garlic-infused oil. Sprinkle half of fontina-mozzarella mixture evenly over oil-brushed dough.
Slide onto pizza stone and bake until golden and crisp, 8 to 12 minutes.
Meanwhile, in a bowl, toss parsley leaves with vinaigrette.
Using peel, remove pizza from oven and slide onto a cutting board. Top with half of parsley salad. Using a vegetable peeler, shave 10 to 12 paper-thin pieces of Parmigiano-Reggiano over top of pizza, then cut pizza into wedges. Repeat with remaining dough and toppings.