How to Make It
Preheat oven to 350°F. Heat 2 tablespoons of the oil in a 9-inch cast-iron skillet over medium. Add onion, and cook, stirring often, until softened, about 6 minutes. Add squash; cover and cook, stirring occasionally, until almost tender, about 8 minutes. Transfer mixture to a medium bowl. Add remaining 3 tablespoons oil to skillet over medium-high. Add Broccolini, mushrooms, and garlic. (Skillet will be very full, but vegetables will cook down.) Cook, stirring occasionally, until almost tender, about 8 minutes. Transfer to bowl with squash mixture. Wipe skillet clean. Let vegetable mixture cool 30 minutes. Add Parmesan, egg, salt, black pepper, and crushed red pepper to squash mixture; stir to combine.
Brush skillet lightly with butter. Fit 1 phyllo sheet into skillet, allowing the edges to hang over the sides. (Cover remaining sheets with a damp paper towel to prevent them from drying out.) Quickly brush sheet lightly with butter, and turn skillet 45 degrees. Continuing to work quickly, top with a second sheet; brush lightly with butter. Repeat with remaining sheets and butter, reserving 1 tablespoon of the butter.
Spoon vegetable mixture evenly over phyllo layers in skillet. Sprinkle goat cheese over top. Fold edges of phyllo sheets up and over filling toward center, overlapping slightly, leaving center exposed. Brush top of phyllo with remaining tablespoon butter. Bake in preheated oven until crust is golden brown, about 40 minutes. Transfer skillet to a wire rack. Cool 15 minutes before serving.