Put 1/4 cup of oil in a medium skillet over medium-high heat and heat until hot but not smoking. Add 1/4 cup of the cheese to a medium bowl and set aside.
Brush a work surface with a little oil and press the dough into a rectangle about 1/4-inch thick. Cut the dough into thin strips, about 1/4-inch wide. When the oil is hot, using kitchen scissors, carefully hold the dough strips, 2 at a time, over the pan and snip off 1/4-inch pieces, letting them gently fall into the hot oil (this may take several batches). Shallow-fry the dough pieces, tossing frequently with a slotted spoon, Chinese strainer or tongs, until golden brown, 3 to 4 minutes. Transfer the croutons to the bowl with the cheese and toss to coat. Continue frying and dredging the dough in the cheese until all the dough is used.
When ready to serve, add the romaine to a large bowl, drizzle with lemon juice and add the remaining 1/4 cup oil. Season with salt and pepper, to taste and toss well to evenly coat the lettuce. Add the remaining 1/4 cup of cheese and the croutons and toss well until combined.
Transfer to a serving platter and top with additional cheese, if desired.