In a medium bowl, mix the warm water, yeast and 1 tablespoon of olive oil. Add the salt and one cup of the flour, stirring with a wooden spoon. Add the remaining flour, a cup at a time, stirring until it all comes together. You can work the dough in the bowl with your hands, turning it over to knead. Or you can turn the dough out onto a lightly floured work surface and knead. Either way, work the dough folding it over and pushing with the heels of your hands until the dough is well mixed, 3-5 minutes. Sprinkle with flour as needed to keep the dough from being too sticky.
Place in an oiled bowl at least twice the volume of the dough. Cover with plastic wrap and let rise at room temperature until doubled in size, about 1 hour. While the dough is rising prepare the onion topping and grate the cheese. After the dough has risen an hour, turn the dough onto the counter and divide into 4 balls. Cover and let these rise for another 15 minutes.
For the topping and grilling:
Cut the onions in half and slice them crosswise about 3/8-inch thick. Trim the shallots and cut them in half lengthwise, then slice thinly. Heat the first 3 tablespoons of olive oil in a large skillet on medium heat. Add the onions and shallots and cook, stirring frequently, until the onions are very soft, have shrunk to half of their original volume and have begun to turn brown on the edges. Add the vinegars, stirring to mix well. Season generously with salt and freshly ground pepper and the thyme leaves.
Pre-heat the grill to high heat. Mix the remaining olive oil with the garlic and set aside. Flatten one dough ball at a time, and on a lightly floured work surface, using a lightly floured rolling pin, roll it into a circle, about 7-inches around and ¼-inch thick. You can pick it up with your hands and gently stretch it. Brush lightly with some of the garlic oil and invert onto a very hot grill. Bake 1 or 2 minutes until there are light grill marks. Turn over and cook the other side a minute. Remove to a clean work surface. When all the crusts are par-baked, lower the grill heat to medium.
Brush the baked crusts with any remaining garlic oil. Sprinkle with a tablespoon or so of the Fontina cheese, and divide the onions between the crusts. Scatter the remaining cheese over the onions and sprinkle with salt and freshly ground pepper. Return the pizzas to the grill. Bake about 5 minutes, to melt the cheese and heat the onions.