In the bowl of an electric stand mixer combine the water, yeast, and sugar. Stir briefly and let stand for 5 minutes or until the top surface has a thin, frothy layer (this indicates that the yeast is active). Add the flour, oil, and salt. Fit the mixer with the dough hook and mix on low speed for about 1 minute or until the dough begins to come together. Increase the speed to medium. Continue to mix until the dough is slightly sticky, smooth, and elastic, about 10 minutes. Form the dough into a ball and place in a lightly oiled bowl. Turn it over to coat all sides and cover the bowl tightly with plastic wrap. Allow the dough to rise in a warm place until it has doubled in size, 1 to 1½ hours.
Brush the cooking grates clean. Grill the bell peppers over direct medium heat, with the lid closed as much as possible, until evenly charred on all sides, 10 to 12 minutes, turning every 3 to 5 minutes. Remove the peppers from the grill and place in a paper bag; close tightly. Let stand 10 to 15 minutes to steam off the skins. Remove the peppers from the bag and peel away the charred skins. Cut off the tops and remove the seeds. Cut each pepper into thin strips.
Brush both sides of the mushroom caps with oil. Season with salt and pepper. Grill over direct medium heat, with the lid closed as much as possible, until tender, 8 to 10 minutes, turning and basting with the oil as needed to keep them moist. Remove from the grill and cut into ½-inch slices.
Place the dough on the cooking grate with the paper sides facing up. Grill over direct medium heat, with the lid closed as much as possible, until the dough is well marked and firm on the underside, 2 to 5 minutes, rotating as needed for even cooking. Peel off and discard the parchment paper. Transfer the crusts to a work surface with the grilled sides facing up.
Grill the pizzas over direct medium heat until the crusts are crisp, 4 to 6 minutes. Remove from the grill. Top with the arugula, tomato, cheese, and lemon juice. Cut into wedges and serve warm.
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