Boil the pasta sheets in salted water until cooked al dente. Drain and set aside.
In an oversize saucepan, bring the whipping cream to a light boil and reduce by about a third. Be careful it doesn't foam over.
Pour about half of the reduced cream into a skillet. Add the garlic, salt, pepper, and crawfish tails. Simmer over low heat while agitating the pan until contents are well combined.
Add the basil and the shredded mozzarella and stir very lightly until the cheese begins to melt. Remove from the heat.
Into the remaining cream, add the Parmesan cheese, white pepper, and reduced crawfish stock for the sauce. Whisk until blended and smooth.
Spoon the crawfish mixture onto the pasta sheets. Roll them into tubes about an inch and a half in diameter. Place the tubes with the seam side down on ovenproof dishes. Spoon about two tablespoons of sauce over each individual cannelloni.
Bake the cannelloni in a 350-degree oven until the sauce begins to bubble, but not so long that the pasta browns. Serve very hot.