Preheat oven to 400ºF. Mist a 9-inch pie plate with cooking spray. Place 2 layers of sweet potato slices over bottom and up sides of pie plate, lightly coating with cooking spray between layers. (You may have some sweet potato left over; cover and refrigerate for another use.) Bake until tender and just beginning to turn golden, about 15 minutes. Remove pie plate from oven and press any sweet potato slices that have slipped down back up sides of dish; reduce oven temperature to 375ºF.
Heat 1 tablespoon oil in a large skillet over medium-high heat. Add onion; cook, stirring occasionally, until tender, 3 to 5 minutes. Add broccoli, remaining 1 tablespoon oil, and, if desired, cayenne; reduce heat to medium. Cover and cook, stirring occasionally, until broccoli is tender, about 10 minutes. (Add a few tablespoons of water to skillet during cooking time if onion is browning too quickly.) Whisk together eggs, milk, salt, and pepper in a bowl until well combined.
Spread broccoli mixture over crust; sprinkle with cheese. Pour egg mixture over cheese. Bake until set, about 25 minutes.