Place a rack in upper third of oven; preheat to 450°F. Drizzle scallions with 1 Tbsp. oil on a small rimmed baking sheet and give them a toss. Roast until softened and charred in spots, 8 to 10 minutes. Let cool; chop.

Coat a large baking sheet with 1 Tbsp. oil. Stretch 1 piece of dough out on baking sheet to form a 16- x 6-inch oval. Bake until golden brown and puffed, 8 to 10 minutes.

While bread is baking, toss mustard greens in a medium bowl with lemon juice and a generous pinch of salt.

Make Olive Oil-Basted Fried Eggs two at a time. For the first batch, heat 3 Tbsp. olive oil in a medium nonstick skillet (a 10-inch skillet is perfect for 2 eggs) over medium-high. When oil is shimmering, carefully add each egg, shaking pan gently after each addition. Cook both eggs, shaking pan occasionally to keep them from sticking to each other (a rubber spatula can help nudge them apart if needed), until edges are golden brown, about 2 minutes. Carefully tilt skillet toward you so that oil pools at front edge of pan. Use a big soup spoon to pour hot oil over egg whites, especially anywhere they are still translucent, while avoiding yolks, until whites are set, about 1 minute more. Transfer to a paper towel–lined plate to blot oil; season with salt and pepper.

Remove flatbread from oven. Coat another baking sheet with remaining 1 Tbsp. oil and bake remaining piece of dough. While remaining dough is baking, spread hot flatbread with 1 oz. nduja, then top with 2 fried eggs. Scatter half of greens over flatbread and top with half of scallions and cheese.

As soon as remaining flatbread comes out of the oven, repeat topping instructions with remaining ingredients.