Make the Dough: Combine the flour, cold butter, sugar, and salt in a bowl, pinching the mixture together with your fingertips until most of the large butter pieces are flattened or broken up and no dry patches of flour remain.
Add the Water: Gradually incorporate enough cold water for the mixture to come together into a rough ball; you may not need the full 1/4 cup.
Form the Dough: Knead the dough only until it starts to become smooth, taking care not to overwork it because excessive handling can make the crust tough.
Chill the Dough: Shape the dough into a ball, flatten it into a disk, wrap it in plastic wrap, and refrigerate for at least 30 minutes; it can also be chilled overnight or frozen for later use.
Cook the Vegetables: Heat a large skillet over medium-high heat, add the olive oil and diced onion, and cook for about 1 minute before adding the chopped broccoli.
Season the Filling: Cook the broccoli and onion for 3–5 minutes, or until the broccoli is tender and bright green, then season with salt and freshly cracked black pepper and set the mixture aside.
Prepare the Oven: Preheat the oven to 375°F (190°C) and line a baking sheet with parchment paper.
Roll the Dough: Transfer the chilled dough to a floured surface and roll it into a circle approximately 12 inches (30 cm) in diameter.
Transfer the Crust: Gently move the rolled dough onto the prepared parchment-lined baking sheet.
Add the Filling: Place the cooked broccoli mixture and grated cheese in the center of the dough, spreading the filling outward while leaving about a 2-inch (5 cm) border around the edge.
Fold the Galette: Fold the uncovered dough edge over the filling, creating creases and overlapping folds as you work around the circle.
Brush the Crust: Brush the exposed crust with the prepared egg wash or heavy cream.
Bake the Galette: Bake at 375°F (190°C) for 50–60 minutes, or until the crust is golden.
Serve: Allow the finished galette to be served warm or at room temperature.