Folding the noodles lengthwise before rolling them up is a clever way to keep the filling from oozing out; the fold creates a cup to hold the filling.
Mix together ricotta, Parmigiano-Reggiano, grated mozzarella, cream cheese, chopped spinach, and garlic. Season with salt and pepper. Stir in flour and egg. Refrigerate while cooking noodles, at least 10 minutes.
Fill a 3 1/2-quart braiser with water. Bring to a boil. Add 1 tablespoon salt and the noodles; cook, stirring occasionally and using tongs to lift the bottom sheets and place on top (for even cooking) until pliable but not fully cooked, 6 to 7 minutes (or boil in a large stockpot). Drain. Run under cold water to stop cooking and rinse off some starch; pat dry. Spread sauce into the bottom of the braiser.
Preheat oven to 400° with one rack in center of oven. Spread each noodle with a generous 3 tablespoons ricotta mixture. Arrange a few spinach leaves on top. Fold in half, lengthwise. Transfer to a parchment-lined baking sheet. Repeat with remaining cheese mixture and noodles. Refrigerate 10 minutes. Starting at one short end, roll up noodles into a spiral, then arrange in pan, ruffled side up.
Tuck torn mozzarella in around rolls. Cover and bake until bubbling around edges, 20 to 25 minutes. Uncover, sprinkle with Parmigiano-Reggiano, and continue baking until bubbling in center and beginning to brown, about 10 minutes. If desired, for additional crispy edges, turn oven to broil, and broil on top rack, until noodles are browning on edges, about 2 minutes. Let stand 10 minutes. Sprinkle with basil just before serving.