Preheat oven to 350 degrees F. Spray a 9-inch pie plate with nonstick cooking spray.
Heat the olive oil in a large, deep skillet over medium-high heat, about 2 minutes. Add the onions and 1/2 teaspoon kosher salt and cook, stirring often, 8 minutes, until lightly browned. Remove from heat and allow to slightly cool.
Meanwhile, in a large bowl, whisk together the eggs, yogurt, 1/2 more teaspoon kosher salt, black pepper, nutmeg and coconut flour. Using a spatula, mix in the cheese and the caramelized onions.
Transfer the mixture to the prepared pie plate. Bake 40-45 minutes, until puffed and set and a toothpick inserted in the center of the quiche comes out clean.
Remove from oven. Cool 10 minutes in the baking dish on a cooling rack, then carefully run a knife around the edges to loosen. Invert onto a cutting board, then back to the cooling rack to finish cooling, about 10 more minutes.
Carefully transfer the quiche back to the cutting board. Use a sharp knife to cut into 6 slices, and serve.