How to Make It
Adjust an oven rack to the lower-middle position and preheat the oven to 300°F.
Place the red onion on a rimmed 10 x 16-inch pan and drizzle with olive oil. Toss with the thyme and spread into an even layer. Cover the sheet pan with aluminum foil (you may need 2 sheets), crimping it tightly around the edges to seal, and roast the onions until they are soft, about 20 minutes. Uncover the pan, stir, and continue to cook until the onions are very soft, sticky, and golden brown, 15 to 20 minutes more. Remove the sheet pan from the oven, sprinkle the onions with 1/2 teaspoon salt, and transfer them to a plate to cool completely (reserve the sheet pan and any grease or seasonings left in it for Step 4).
Increase the oven temperature to 400°F.
Lightly flour your work surface and put the puff pastry on top. Lightly roll out the pastry to between 1⁄8 and 1/4 inch thick. Transfer the pastry to the reserved sheet pan—it should just cover the bottom. With a fork, prick the dough all over, then place the sheet pan in the refrigerator while you make the filling.
Whisk the whole eggs and egg yolks to combine in a large bowl. Whisk in the milk and cream, and add the remaining 1/2 teaspoon salt, and the pepper. Remove the crust from the refrigerator and sprinkle half the Gruyère over it. Add the onions in an even layer and then the spinach followed by the remaining Gruyère. Pour the egg filling over and crumble the goat cheese over the top.
Bake the quiche for 10 minutes. Reduce the oven temperature to 375°F and continue to bake until the filling is set around the edges and bounces back to light pressure in the center, 20 to 25 minutes.
Turn the broiler on to high and adjust an oven rack to the top position. Move the quiche to the top rack and broil until browned, about 1 minute (watch the quiche closely as broiler intensities vary). Remove from the oven, slice, and serve warm or at room temperature.