In an 8- to 10-inch frying pan over high heat, stir oil and ginger until ginger is golden brown, 2 to 4 minutes. With a slotted spoon, transfer ginger to paper towels. Let oil cool.
With a vegetable peeler, shave about half the cheese into very thin slices. Grate remaining cheese.
In a wide, shallow bowl, whisk oil, grated cheese, garlic, and lemon juice to blend.
Break romaine into bite-size pieces into bowl. Add croutons and mix gently.
Sprinkle fried ginger and shaved parmesan over salad.