Fry the bacon in a skillet until crisp. Set aside on a paper-towel lined plate. When cool enough to handle, crumble.
While bacon is cooling, use a vegetable peeler to shave the asparagus into thin strips. (The best way to do this is to place the asparagus flat on a cutting board and run the peeler over the top, rotating the asparagus as you peel. If you have a spiralizer, you can also use that to spiralize the asparagus.) Toss the asparagus with olive oil and a pinch of salt.
Turn on the oven's broiler (set it to its lowest setting if possible). Place an oven rack about 6 inches from the broiler.
Spray a sheet pan with nonstick cooking spray or line it with parchment paper.
Set the flatbreads face-down on the sheet pan and broil until they are lightly toasted on the bottom. (Toasting the bottom will help prevent the flatbread from turning soggy when you pile it with the other ingredients.)
Flip the flatbreads face-up and spread with pesto. Top with the mozzarella and then the asparagus (don't worry if every inch of the flatbreads aren't covered with mozzarella; these are much more beautiful if some of the asparagus is peeking out).
Place the flatbreads under the broiler and broil until the cheese is melted and the asparagus is starting to brown in spots. Keep a close eye on them to prevent burning.
Remove flatbreads from the oven and slice into wedges. Transfer to a serving platter and top with crumbled bacon and parmesan cheese.