As a note, make the dressing first! You can make this a day or two ahead of time and store it in the fridge too.
Preheat your grill to a medium-high setting. Let the grates get hot for at least 10 minutes!
Remove the woody stems from the asparagus spears. Toss them with 1.5 tablespoons of the olive oil and season all over with salt and pepper.
Brush the sourdough slices with the remaining olive oil. Sprinkle all over with kosher salt and even some garlic powder.
Place the asparagus spears directly on the grill (in the opposite direction of how the grates run, so they don’t fall through the cracks!). Grill for 5 to 6 minutes, tossing occasionally, until the spears are slightly charred. I also throw the sourdough on the grates at the same time. The bread needs about 1 to 2 minutes per side, until it’s toasty with grill marks.
Remove the asparagus and bread from the grill. Slice or tear the sourdough into smaller crouton pieces.
Place the asparagus on a plate and drizzle all over with the caesar dressing - use as much or as little as you’d like. Top with the shaved parmesan cheese. Top with the sourdough croutons. Top with the finely grated parmesan. Serve immediately!