In a large skillet over medium-high heat, warm 3 tablespoons of the oil. Add the onions and sugar and cook until golden, about 15 minutes, stirring occasionally. Add the vinegar and cook for 1 minute. Remove from the heat and stir in the salt. Transfer the onions to a bowl and return the skillet over medium-high heat. Warm 2 teaspoons of the remaining oil and add the bell pepper. Cook until crisp-tender, 3 to 4 minutes, stirring occasionally. Remove from the heat.
Divide the dough into four equal pieces. Lightly brush four 9-inch squares of parchment paper on one side with oil. Using your fingers, flatten each piece of dough on an oiled square of parchment paper to create four rounds, each 7 to 8 inches in diameter and about ¼ inch thick. Lightly brush the tops of the dough with oil. Allow the dough to stand at room temperature for 10 minutes.
Prepare the grill for direct cooking over medium heat (350°F to 450°F).
Brush the cooking grates clean. Grill the dough, paper side facing up, over direct medium heat, with the lid closed, until it is well marked and firm on the underside, 2 to 5 minutes, rotating as needed for even cooking. Discard the parchment paper. Transfer the crusts to a work surface with the grilled side facing up.
Evenly divide the mozzarella, onions, bell peppers, blue cheese, and walnuts over the crusts. Return the pizzas to the grill and cook over direct medium heat, with the lid closed, until the cheese is melted and the bottom of the crusts are crisp, 2 to 5 minutes, rotating the pizzas occasionally for even cooking. Transfer to a cutting board, sprinkle with parsley, and cut into wedges. Serve warm.
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