Heat the oven to 400°F.

• Peel and thinly slice the red onion.

In a medium frying pan over medium-high heat, warm 2 tablespoons oil until hot but not smoking. Add the onion, season with salt and pepper, and cook, stirring occasionally, until softened and starting to caramelize, 8 to 10 minutes.
While the onion cooks, start the flatbreads.

• Crack the eggs into separate small cups or mugs.

On a sheet pan, place the flatbreads side by side and brush the tops lightly with oil. Spread the onion evenly over the flatbreads, leaving a ½-inch border around the edges, and top with about one-third of the arugula (set aside the remaining two-thirds for the salad). Slide 1 egg onto each flatbread, season with salt and pepper, and sprinkle with the goat cheese.

Bake until the crust is crisp, the egg whites have set, and the yolks are slightly runny, 12 to 15 minutes.
When the flatbreads are almost done, prepare the basil-lemon dressing and arugula salad

• Juice the lemon.
• Strip the basil leaves from the stems; finely chop the leaves.

In a small bowl, stir together 1 tablespoon lemon juice, basil, and 1 to 2 tablespoons oil. Season the dressing to taste with salt and pepper.
In a medium bowl, toss the remaining arugula with as much dressing as you like. Set aside any remaining dressing for the flatbreads.

Transfer the flatbreads to individual plates and drizzle with any remaining dressing. Top with the arugula salad and serve.