Adjust an oven rack to the lower-middle position, place a baking stone on the rack, and heat the oven to 500°F. Let the baking stone heat for at least 30 minutes or up to 1 hour to ensure the base is hot enough to crisp the dough.

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

Remove the lid and increase the heat to medium-high. Continue to cook, stirring often, until the moisture evaporates and the onions are deeply browned and jammy, about 10 to 15 minutes.

Transfer the dough to a lightly floured counter and divide it into two equal pieces. Cover the dough with greased plastic wrap and let it rest for 10 minutes so the gluten relaxes, making it easier to roll.

Work with one piece of dough at a time, keeping the other covered. Press and roll the dough into a 14 by 8-inch oval. Transfer the dough to a parchment paper-lined rimless baking sheet and gently dimple the surface with your fingertips.

Brush the dough liberally with oil and season with pepper. Scatter half of the caramelized onions, half of the walnuts, and half of the blue cheese evenly over the dough, leaving a 1/2-inch border around the edge.

Slide the parchment paper and flatbread onto the hot baking stone. Bake until the crust is deep golden brown and the cheese is melted, about 10 minutes, rotating the flatbread halfway through to ensure even browning.

Remove the flatbread by sliding the parchment back onto the baking sheet. Transfer to a cutting board, discard the parchment, and sprinkle with half of the scallions. Slice and serve while the cheese is still warm and the crust is crisp.