Move oven rack to upper-middle position and preheat oven to 375 degrees F.
COOK SHELLS:
Cook shells as per package instructions until al dente. Drain shells and transfer to kitchen towel-lined baking sheet. Reserve 24 shells, discarding any that have broken.
MAKE MEAT SAUCE:
While shells cook, make meat sauce. In large skillet over medium heat add ground beef and cook until browned. Drain off fat and add sauce. Mix until combined.
Spread half meat sauce into bottom or 13 x 9-inch baking dish (or any other, large 3-quart pan); set aside.
PREPARE FILLING:
Pulse crackers in food processor until finely ground. Add cottage cheese, 1 1/2 cups mozzarella, Parmesan, basil, garlic, and 1/2 teaspoon salt and process until smooth.
Place mixture into quart-sized plastic bag, seal and snip off 1 corner of bag. Pipe 2 tablespoons filling into each shell.
ASSEMBLE:
Nestle the filled shells, seam side up, on top of meat sauce. Evenly distribute remaining sauce over shells.
Cover the pan with aluminum foil and bake until bubbling around edges, 30-35 minutes. Remove foil and sprinkle with remaining mozzarella.
Bake until cheese is just melted, about 5 minutes.
Let cool for 10 minutes. Garnish with additional Parmesan cheese and basil. Serve.