Preheat the oven to 400°F. Line a large baking sheet with parchment paper and set aside.
Peel and thinly slice the onions and set them aside.
Trim the asparagus, rinse, and pat them dry. Set aside.
In a large skillet, melt two tablespoons of butter over medium-high heat with two tablespoons of EVOO.
Add the sliced onions and cook, stirring often, until soft and golden brown (about 10–15 minutes)—season with salt and pepper to taste. Set aside.
Combine both dough portions into one ball, dust with two tablespoons of all-purpose flour, and gently knead. Shape into a large oblong and transfer to the parchment-lined baking sheet.
Drizzle a little EVOO over the surface of the dough and use your fingers to spread it evenly. Top with sautésautéedns, then arrange the asparagus spears evenly over the top. Finish with the two cups of shredded Toscano cheese.
Place the flatbread on the center rack of your oven and bake for 15–20 minutes, or until the crust is golden brown and the cheese is melted and bubbly.
Remove from the oven. Slice and serve immediately. Optional: Drizzle with a little more olive oil.