In a large nonstick skillet, melt the butter. Add the squash and 1 cup water and simmer over medium heat; season with salt and pepper. Cook, stirring, until the water is evaporated and the squash is golden and tender, about 13 minutes; set aside.
Position a rack in the upper third of the oven and preheat to 400 degrees . Meanwhile, lightly grease an 8-inch square glass baking dish and spread 1/2 cup white sauce in the bottom. In a bowl, whisk together the pumpkin puree and egg.
Cut half of the egg roll wrappers into 3 strips each. Fill a large bowl with hot tap water (as you work with the wrappers, dip them in the water to rinse off the starch). Use 1 whole wrapper and 3 strips to cover the bottom of the dish in a single layer (there will be overlap). Spread a generous 1/2 cup of pumpkin mixture on top and sprinkle with 1/3 cup cheese. Repeat the layering.
Add another layer of egg roll wrappers. Spread with 1 cup white sauce, then top with the squash, basil and 1/3 cup cheese. Add another layer of egg roll wrappers, the remaining pumpkin mixture and 1/3 cup cheese. Top with the remaining egg roll wrappers, then spread with the remaining white sauce.
Cover the dish snugly with foil and bake on a baking sheet in the upper third of the oven for 30 minutes. Uncover the dish, sprinkle the remaining 3/4 cup cheese on top and bake until golden and bubbling, about 10 minutes. Let cool for 15 minutes before cutting.