1. To prepare ravioli, place ricotta in a cheesecloth-lined colander; drain 30 minutes. Combine ricotta, 1/4 cup Parmigiano-Reggiano, and next 4 ingredients (through egg), stirring until well combined.
2. Place 1 (15 x 3–inch) Classic Pasta Dough sheet on a lightly floured surface. Spoon 1 1/2 teaspoons filling mixture 1 1/2 inches from left edge in the center of sheet. Spoon 1 1/2 teaspoons filling mixture at 3-inch intervals along the length of sheet. Moisten edges and in between each filling portion with water; place 1 (15 x 3–inch) pasta sheet on top, pressing to seal. Cut pasta sheet crosswise into 5 (3 x 3–inch) ravioli, trimming edges with a sharp knife or pastry wheel. Place ravioli on a lightly floured baking sheet (cover with a damp towel to prevent drying). Repeat procedure with remaining pasta sheets and filling mixture to form 20 ravioli.
3. Bring 6 quarts water and salt to a boil in an 8-quart pot. Add half of ravioli to pot; cook 1 1/2 minutes or until no longer translucent. Remove ravioli from water with a slotted spoon. Repeat procedure with remaining ravioli.
4. To prepare sauce, heat oil in a large skillet over low heat. Add garlic to pan; cook 6 minutes or until garlic is tender. Remove from heat. Place 5 ravioli in each of 4 shallow bowls; drizzle each serving with 1 1/2 teaspoons garlic oil. Top each serving with 1 tablespoon basil and 1 tablespoon shaved Parmigiano-Reggiano. Serve immediately.