In a large skillet over medium-high heat on the stove, warm 3 tablespoons oil. Add the onions and sugar and cook until golden, about 15 minutes, stirring occasionally. Add the vinegar and cook for 1 minute more. 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 oil and add the bell pepper. Cook until crisp-tender, 3 to 4 minutes, stirring occasionally. Remove from the heat. Add to the onions and allow to completely cool.
Divide the dough into four equal pieces. Lightly brush four 23-centimeter 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 17 to 20 centimeters in diameter and about 6 millimeters 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 (180° to 230°C).
Prepare the grill for cooking over medium heat (200°C).
Brush the cooking grates clean. Grill the dough, paper side facing up, over direct medium heat, with the lid closed, until 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.
Brush the cooking grates clean. Grill the dough, paper side facing up, over medium heat, with the lid closed, until 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, onion and bell pepper mixture, 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, top with parsley, and cut into wedges. Serve warm.
Evenly divide the mozzarella, onion and bell pepper mixture, blue cheese, and walnuts over the crusts. Return the pizzas to the grill and cook over 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, top with parsley, and cut into wedges. Serve warm.
