Adjust an oven rack to the lowest position, place a baking stone on the rack, and heat the oven to 500°F. If using a rimless baking sheet instead of a stone, invert it on the rack before heating.

Heat the oil in a large skillet over medium-low heat. Add the onions, brown sugar, and a pinch of salt. Cover and cook, stirring occasionally, until the onions have released their moisture and softened.

Remove the lid and increase the heat to medium-high. Continue to cook, stirring frequently and scraping up any browned bits from the bottom of the pan, until the onions are deeply browned and jammy. Season with salt and pepper to taste, then let them cool slightly.

Divide the pizza dough into two equal pieces. On a lightly floured piece of parchment paper, roll each piece into a 14 by 8-inch rectangle. The dough should be thin but even.

Brush the edges of the dough lightly with oil. Spread the caramelized onions evenly over the surface, leaving a half-inch border around the edges. Sprinkle the blue cheese and walnuts over the top.

Slide the parchment with the flatbreads onto the preheated stone. Bake until the crust is deeply browned and crisp around the edges.

Transfer the flatbreads to a wire rack to prevent the bottom from steaming. Garnish with scallions and serve warm while the cheese is still soft.