For the dough, combine the water, yeast, honey, and 3 tablespoons of olive oil in the bowl of an electric mixer fitted with the dough hook. Add 3 cups of the flour, then 2 teaspoons of salt and mix for about 10 minutes, slowly adding up to 1 more cup of flour to just keep the dough from sticking to the bowl.
When the dough is ready, turn it out onto a floured board and knead by hand a dozen times. It should be smooth and elastic.
Brush a medium bowl with olive oil, place the dough in the bowl, and turn it several times to cover lightly with oil. Cover the bowl with a clean, damp kitchen towel and set aside at room temperature for 30 minutes, until doubled in size.
Divide the dough into 4 equal parts and roll each one into a smooth ball. Place the balls on a sheet pan and cover them with the damp towel. Allow the dough to rest for 10 minutes.
If you’ve chilled the dough, leave it at room temperature for 30 minutes to allow it to come to room temperature.
To make the pizza, preheat the oven to 500 degrees F.
Take a quarter of the pizza dough, 1 ball, and roll into a 10-inch circle. Sprinkle a sheet pan with the cornmeal and transfer the dough to the pan. (The remaining dough can be refrigerated for up to 2 days.)
To make the sauce, add the tomatoes, olive oil, and salt to a blender and blend until smooth. Spread 1/4 to 1/2 cup of the tomato sauce onto the dough, leaving a 1/2-inch border. Sprinkle over the mozzarella cheese on top of the sauce. On half of the pizza, crumble over 1 cube of marinated sheep and goat cheese. Tear the basil leaves over all of the pizza. Cook for 12 minutes, until the dough is crisp and the mozzarella is melted.
While the pizza bakes, make the arugula topping. Whisk together 1/2 cup olive oil, 1/4 cup freshly squeezed lemon juice, 1 teaspoon salt and 1/2 teaspoon pepper. Place 8 ounces baby arugula in a large bowl and toss with enough vinaigrette to moisten.
Remove the pizza from the oven and top half of the pizza with the arugula salad topping.
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