Finely chop leaves from 1 sage sprig to equal about 1 tablespoon. Combine chopped sage and butter in a medium skillet over medium. Cook, swirling occasionally, until butter is lightly browned and fragrant, 4 to 5 minutes. Remove half of butter-sage mixture from skillet; combine with 2 tablespoons of the oil in a small bowl, and set aside. Remove skillet from heat, and add pumpkin, Parmigiano-Reggiano, lemon juice, and salt to skillet, stirring to combine. 

Place 8 wonton wrappers on a cutting board. Brush outside edges with some of the beaten egg. Place 1 rounded tablespoon of pumpkin filling in center of each wrapper, leaving a 1⁄2-inch border around the edge. Place a second wrapper on top, pressing the edges together to seal. Press gently on filling to spread evenly. Move ravioli to a tray, and cover with plastic wrap. Repeat procedure with remaining wrappers, beaten egg, and pumpkin mixture. 

Remove leaves from remaining sage sprig. Heat remaining 1 tablespoon oil in a small skillet over medium-high. Add sage leaves to skillet; cook until crispy, 2 to 3 seconds. Remove from skillet, and set aside. 

Bring a large pot of salted water to a boil over high. Add half of the ravioli to pot, adding 1 at a time. Cook until ravioli are tender, 1 to 2 minutes. Remove from water with a slotted spoon, and drain excess water. Transfer cooked ravioli to a plate. Repeat procedure with remaining ravioli. Divide cooked ravioli evenly among 4 plates, and drizzle with reserved sage brown butter-and-oil mixture. Garnish servings with crispy sage leaves.