Preheat oven to 400 degrees F. Cut squash in half, scoop out the seeds and place face side down on a baking sheet lined with unbleached parchment paper. Poke skin randomly with a fork and bake for 30 minutes. Remove from the oven and set aside to cool until safe to handle.
Using a fork, separate spaghetti squash into strands and add to a medium bowl. Sprinkle with 1/4 tsp salt and mix. Using your hands, squeeze out liquid a bit. Return to the bowl along with 1/2 tsp basil and 1/2 tsp garlic powder; mix to combine. Spray deep pie dish with cooking spray and transfer squash into it. Press evenly on the bottom and sides using your hands. Set aside.
Preheat ceramic non-stick skillet on medium heat and swirl oil to coat. Add onion and bell pepper; sauté for 5 minutes or until golden brown, stirring occasionally.
In the meanwhile, add eggs, balsamic vinegar, 1 tsp basil and 1/4 tsp salt into previously used bowl and whisk. Add hard cheese, feta cheese, previously sautéed and fresh vegetables; stir well to mix. Pour into prepared crust and flatten with spatula
Bake in 400 degrees F oven for 50 mins or until the knife inserted comes out clean. Let quiche cool down for at least 30 minutes. Cut into 8 slices and serve hot or cold.
Operating without a traditional flour-based pastry shell requires managing moisture aggressively, because roasted squash naturally retains significant water that will ruin an egg custard. Roasting the squash halves face-down at 400 degrees Fahrenheit for 30 minutes concentrates the flesh while allowing steam to escape through the pierced skin. Once cooled, shredding the strands and physically squeezing out excess liquid prevents the filling from becoming watery and dense during baking. Pressing this fibrous vegetable matrix firmly into a greased pie dish creates a sturdy, grain-free crust capable of holding a heavy liquid center. Sauteing onions and bell peppers in a ceramic skillet for 5 minutes drives off surface moisture while caramelizing natural sugars, concentrating flavor before the vegetables join the cheese and eggs. Balsamic vinegar adds acidity that cuts through the rich profile of the hard cheese and crumbled feta without thinning the batter. Whisking the eggs thoroughly incorporates air to lighten the custard, which solidifies as proteins coagulate around 400 degrees Fahrenheit during the 50-minute oven phase. Allowing the finished dish to rest undisturbed for 30 minutes permits the internal starches and proteins to set completely, ensuring clean slices rather than a collapsed wedge. Doubling or halving this formulation requires monitoring the internal baking duration rather than the temperature, as thicker volume demands extended heat penetration to reach the center.