For the crust, add the butter, salt, pumpkin seeds, cheddar cheese, and flour to the bowl of a food processor. Pulse for 1 to 2 minutes until combined. Slowly pour the water while pulsing until the dough clumps around the blade.
If you don’t have a food processor, massage the butter in the dry ingredients using your hands until the dough reaches a clumpy, wet-sand-like consistency. Pour in the water and mix to incorporate.
Dump the dough on the work surface. Press the clumps of dough together to form a rough, shaggy dough. Wrap the dough in plastic wrap. Roughly shape it into a thick disk. Chill for at least 30 minutes in the fridge, up to overnight.
The dough freezes very well, so feel free to make it weeks ahead.
In the meantime, heat the oven to 400°F or 205°C. Line a large baking sheet with parchment paper.
Combine all filling ingredients in a large mixing bowl: thinly sliced squash, thinly sliced shallots, minced garlic, extra-virgin olive oil, chopped rosemary leaves, pumpkin seeds (if using), black pepper, cayenne pepper, ground coriander seeds, kosher salt, vinegar or lemon juice, extra-sharp cheddar cheese, and all-purpose flour. Toss everything together.
Divide the cold dough into thirds. Stack the three pieces of dough on top of each other. Lightly flour the work surface and rolling pin. Roughly shape the dough into a ball. Roll the cold dough to a 0.5 cm thickness, maintaining a circle shape.
The final dough should measure 14 inches (36 cm) in diameter.
Lift the dough and place it on the lined baking sheet. Dump the filling in the middle of the dough, piling the squash in the center, reserving a 3-inch (8 cm) rim around the dough for folding over the filling.
Fold the reserved 3-inch (8 cm) rim over the filling. Beat the egg with three tablespoons of water. Brush the edge with egg wash. Sprinkle flaky salt on top.
Place in the oven on the middle rack. Bake for 45 to 65 minutes, until the edges of the galette are golden brown and hard to the touch.
Make sure to check on the galette halfway through baking. If the edges are browning too fast, cover the edges with aluminum foil, exposing the filling in the center.
Slice and serve.
