Place a baking stone on the lower-middle oven rack and heat the oven to 500°F. Allow the stone to heat for at least 30 minutes to ensure it is hot enough to crisp the dough instantly.

Heat the oil in a 12-inch nonstick skillet over medium-low heat until shimmering. Stir in the onions, brown sugar, and salt, then cover and cook until the onions have softened and released their juices, about 10 minutes.

Remove the lid and increase the heat to medium-high. Cook for 10 to 15 minutes, stirring often, until the onions are deeply browned and caramelized.

Divide the dough in half on a lightly floured counter and cover with greased plastic wrap. Working with one piece at a time, roll the dough into a 14 by 8-inch oval, then transfer to a parchment-lined rimless or inverted baking sheet.

Gently dimple the surface of the dough with your fingertips. Brush the dough liberally with oil and season with pepper, then scatter half of the caramelized onions, walnuts, and blue cheese evenly over the surface, leaving a half-inch border around the edge.

Slide the parchment paper and flatbread onto the hot baking stone. Bake for about 10 minutes, rotating the flatbread halfway through, until the crust is deep golden brown.

Remove the flatbread from the oven by sliding the parchment back onto the baking sheet, then transfer it to a cutting board. Sprinkle with scallions and serve immediately while the crust is still crisp.