Preheat convection oven to 350°F or conventional oven to 375°F. Line 4 full sheet pans with parchment paper. Coat two 2-inch full hotel pans with cooking spray.
Cut potatoes into 1⁄8-inch slices using a food processor fitted with a slicing blade or by hand.
Peel sweet potatoes. Cut into 1⁄8-inch slices using a food processor fi tted with a slicing blade or by hand.
Peel, quarter and seed squash. Cut into 1⁄8-inch slices using a food processor fi tted with a slicing blade or by hand.
Toss the potatoes, sweet potatoes and squash with ¼ cup oil, water and salt. Divide between the prepared sheet pans and spread in an even layer. Bake until the vegetables are tender, 10 to 15 minutes. Let cool to room temperature.
Heat the remaining 1 cup oil in a 7-qt saucepan over medium heat until slightly bubbling. Whisking constantly, gradually sprinkle fl our over the oil and whisk until smooth. Cook, whisking constantly, for 5 minutes. Do not let burn. Whisking constantly, slowly add milk. Cook until slightly thickened, about 15 minutes.
Combine breadcrumbs and cheese in a small bowl.
Layer the vegetables in the prepared hotel pans. Pour half of the sauce evenly over each pan of vegetables. Sprinkle each with half of the breadcrumb mixture.
Bake until golden brown and a knife easily cuts through the layers, about 30 minutes. Let cool slightly and use a 6-oz spoodle or #6 disher to serve.
