Spray one 9" pie pan and set aside. Preheat the oven to 350 degrees F. To make the crust, combine flour and salt in a bowl. Add non-dairy butter in small chunks and with a pastry cutter or your fingers, cut into the dough until it resembles small pea-sized pieces (a few larger ones are okay). Add cold water and turn over onto a floured surface to bring together. It will be pretty dry but should stick together when pressed between your fingers.
Press into the prepared pie pan. Poke holes in the bottom of the crust with your fork and chill the crust for 10 minutes. Once chilled, remove from refrigerator and bake for 10 minutes. Set aside to cool while making the filling.
To make the quiche filling, heat 1 tablespoon of oil over low heat. Add sliced onions and sauté to caramelize until golden and fragrant (this takes about 10 minutes). Add garlic and shiitake mushrooms and sauté another 5 minutes, stirring frequently. Once mushrooms are golden brown, add lettuce and cook for 1 minute until lettuce is wilted. Remove from heat and set aside.
In a food processor or blender combine tofu, nutritional yeast, 2 tablespoons non-dairy milk, flour, onions, lemon juice, sea salt, and turmeric (if using) until smooth, scraping down the sides as necessary. You may need to add an additional tablespoon of milk to get your mixture smooth. Transfer to a bowl. Add mushroom/onion mixture and fresh basil. Stir everything together until vegetables are evenly incorporated. Season with pepper and adjust seasonings to taste.
Spoon filling into chilled pie crust, smoothing over the top. Bake for 40-45 minutes until the top has browned slightly and the quiche is firm when wiggled.
Remove from the oven and allow the quiche to cool for 20 minutes before slicing. Serve warm.