1 ¼cupall-purpose flourplus a little more for rolling out the dough
8tbspcold unsalted butter
1tbspdistilled white vinegar
For the filling
1cupgrated Italian fontina cheese
1eggfor egg wash
Make the dough
Place the flour and salt in the bowl of a food processor. Pulse once to combine.
Cut the cold butter into cubes. Add to the food processor and pulse till pieces of butter are the size of very small peas, about 10 pulses
With the motor of the food processor running, pour in the water and vinegar. Pulse until it just starts to come together.
Dump the mixture onto a piece of plastic wrap. Using the edges of the plastic wrap, press the dough together to form a ball. Cover with the plastic and form into a disc. Refrigerate for 1 hour.
Make the filling
While the dough is resting, heat the oven to 375°F. Peel the squash and cut it into ½ inch pieces. Spread out onto a sheet pan (line with a silicone mat or aluminum foil for easy cleanup) and toss with olive oil and salt. Roast for about 45 minutes, or until very soft. When squash is done, turn the heat up to 400°F.
While the squash is roasting, melt butter in a pan and add the sliced onion. Sprinkle with a little salt. Add about two tablespoons of water, cover, and cook on low heat till onions are softened and translucent. Remove the lid and continue to cook over medium heat, stirring occasionally, until onions have caramelized to a light brown.
Assemble and bake the galette
Remove the dough from the fridge. Sprinkle your work surface with some flour and roll out the dough to about a 13" circle. Transfer the dough to a parchment-lined sheet pan.
Scatter the grated fontina on top of the dough, leaving a border of about 1 ½ - 2 inches. Top with the caramelized onion and the cooked squash.
Fold the edges of the dough up and over the filling, folding and pinching the dough as you go. The center of the galette will remain open.
In a small bowl, mix one egg with about 1 tbsp of water. Brush this mixture carefully over the exposed edges of the dough. (you will not use all of the egg)
Place in the 400°F oven and bake until the crust is golden brown, rotating the sheet pan halfway through for even cooking. Let rest for about 5 minutes before cutting. Cut into 4 or 6 pieces.