In a small bowl, mix the warm water, yeast, olive oil and honey.

In the work bowl of a heavy-duty electric mixer, or in a large bowl, mix 1 1/2 cups of the flour and the salt. Add the yeast mixture and stir with a wooden spoon until it all comes together.

If using a machine, fit the bowl with a dough hook and knead on a low speed. Otherwise, turn the it out onto a lightly floured surface and knead by hand. Knead 4 to 5 minutes, adding the remaining 1/4 cup of flour, and a little more, if necessary, until the dough is soft, smooth and elastic.

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 topping ingredients and preheat the oven to 475 degrees. If you have a baking stone, place it on the middle oven rack.

Divide the dough in 2 portions and roll the first on a floured surface to a 13-inch round. If you are using a stone, sprinkle a wooden pizza peel or rimless baking sheet with cornmeal. Otherwise, sprinkle a round baking pan with cornmeal. Transfer the crust to the wooden peel or the baking pan. Brush generously with some olive oil.

Scatter half of the cooked mushrooms as evenly as you can, leaving a 1-inch border. Sprinkle with half of the goat cheese and then half of the walnuts. If you are using the wooden peel, transfer the pizza to the preheated baking stone, using the cornmeal to release the crust onto the stone. Bake until the crust is brown, less than 10 minutes on the stone—add a few more minutes on the baking pan. Immediately scatter half of the dressed arugula over the baked pizza and slide onto a serving platter.