Preheat oven to 425 degrees F. Line 2 baking sheets with parchment paper.

In a large bowl, use a fork to whisk together the whole-wheat flour, all-purpose flour, and salt. Drizzle the olive oil over the flour and mix until crumbly. Make a well in the center and pour in the water and vinegar. Mix with a fork until the dough comes together. Put the dough on your work surface and knead several times until smooth and pliable without being sticky. Wrap with plastic wrap and rest at room temperature for 1 hour.

While the dough rests, arrange the mushrooms, squash, and shallots on the baking sheets. Drizzle the olive oil over the top, season with ½ teaspoon salt, and a light shower of black pepper. Toss everything together. Roast for 15 minutes, stir, and continue to roast until you can slide the tip of a knife into the squash with relative ease, 30 to 35 minutes (if the shallots begin to burn before then, remove from the baking sheet).

Scatter the kale over the vegetables and continue to roast everything for another 5 minutes, so the kale wilts.

Remove vegetables from oven and lower the temperature to 350℉.

Sprinkle a little flour on your work surface. Use a rolling pin to roll out the dough. If you find it sticking quite a bit, add a little more flour to both your work surface and the top of the dough. Roll it out until it’s big enough to fit your pie pan with ½ inch draping over the sides (about 12 inches in diameter). Transfer it to a pie pan and pinch it along the rim to form a crust.

In a large bowl, whisk the eggs, milk, cottage cheese, and remaining salt and pepper until creamy and smooth. Add 1 cup of the Gruyere and whisk again.

To assemble, transfer the vegetables to the pie pan, making sure the squash, mushrooms, and kale are evenly distributed. Pour the egg mixture over the top and sprinkle with the remaining cheese.

Bake at 350 degrees until the crust is nicely browned and the center is just set, 40 to 50 minutes. If desired, set it under the broiler for about 30 seconds to brown the surface.

Leave the pie to rest for 15 minutes before cutting into wedges.