For the dough, combine the water, yeast, honey, and 3 tablespoons olive oil in the bowl of an electric mixer fitted with a dough hook. When the yeast is dissolved, add 3 cups of the flour, then 2 teaspoons salt, and mix on medium-low speed. While mixing, add up to 1 more cup of flour, or just enough to make a soft dough. Knead the dough for about 10 minutes until smooth, sprinkling it with the flour as necessary to keep it from sticking to the bowl. When the dough is ready, turn it out onto a floured board and knead it by hand a dozen times. It should be smooth and elastic. Place the dough in a well-oiled bowl and turn it to cover it lightly with oil. Cover the bowl with a kitchen towel and allow the dough to rise at room temperature for 30 minutes.
Meanwhile, make the garlic oil. Place 1/2 cup olive oil, the garlic, thyme, and red pepper flakes in a small saucepan and bring to a simmer over low heat. Cook for 10 minutes, making sure the garlic doesn't burn. Set aside.
Preheat the oven to 500 degrees. (Be sure your oven is clean!)
Dump the dough onto a board and divide it into 6 equal pieces. Place them on a sheet pan lined with parchment paper and cover them with a damp towel. Allow the dough to rest for 10 minutes. Use immediately, or refrigerate for up to 4 hours.
Press and stretch each ball into an 8-inch circle and place 2 circles on each parchment-lined sheet pan. (If you've chilled the dough, take it out of the refrigerator approximately 30 minutes ahead to let it come to room temperature.) Brush the pizzas with the garlic oil, and sprinkle each one liberally with salt and pepper. Sprinkle the pizzas evenly with Fontina, mozzarella, and goat cheese. Drizzle each pizza with 1 tablespoon more of the garlic oil and bake for 10 to 15 minutes, until the crusts are crisp and the cheeses begin to brown.
Meanwhile, for the vinaigrette, whisk together 1/2 cup of the olive oil, the lemon juice, 1 teaspoon salt, and 1/2 teaspoon pepper. When the pizzas are done, place the arugula in a large bowl and toss with just enough lemon vinaigrette to moisten. Place a large bunch of arugula on each pizza and serve immediately.
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