Preheat an outdoor grill or cast-iron grill pan to medium-high heat (approximately 400°F/204°C).

Toss the trimmed asparagus spears with olive oil, salt, and pepper in a mixing bowl until uniformly coated.

Arrange asparagus on the grill grate perpendicular to the bars to prevent falling through.

Grill the asparagus for 4 to 6 minutes, rotating occasionally until distinct grill marks appear and the spears are tender-crisp.

Remove asparagus from heat and allow to cool for 2 minutes; optionally cut into 2-inch segments.

In a large chilled mixing bowl, combine the chopped romaine lettuce with the Caesar dressing and toss using tongs until the leaves are evenly emulsified.

Incorporate the sourdough croutons and half of the shaved Parmesan cheese into the lettuce mixture.

Apportion the salad into four chilled bowls and top each with an equal portion of the grilled asparagus spears.

Garnish with the remaining Parmesan cheese and serve immediately with lemon wedges for acidity.