While the crust is baking, mix 1 large sweet onion (such as Vidalia), thinly sliced, 1 Tbsp. extra-virgin olive oil, and a large pinch of kosher salt in a medium skillet. Cover pan and cook onion over medium heat, stirring occasionally, until softened and translucent, 8–10 minutes. Uncover and cook, adding splashes of water (up to ¼ cup) and stirring and scraping bottom of pan often, until deep golden brown, 12–15 minutes.
Whisk 3 large eggs, 3 sprigs thyme, leaves finely chopped, ½ cup heavy cream, ½ cup whole milk, ½ tsp. freshly ground pepper, and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt in a medium bowl. Toss together 2 oz. Gruyère, finely grated, and 1 oz. Parmesan, finely grated, in a small bowl. Spread caramelized onion evenly over crust. Sprinkle half of cheese on top. Scatter 2 cups baby spinach, then remaining cheese over. Pour in egg mixture and lightly press spinach down with a spoon to submerge.
Bake quiche until edges of filling are set but center still wobbles slightly when gently shaken, 45–50 minutes. Transfer to a wire rack; let cool 1 hour before slicing. Do Ahead: Quiche can be baked 2 days ahead. Let cool completely; tightly wrap and chill.
