Preheat the oven to 180°C (350°F).
Heat the butter in an ovenproof or cast-iron skillet over medium heat, then add the olive oil.
Add the sliced red onion and sliced garlic to the skillet and cook over medium-low heat until the onions are golden-brown and beginning to caramelize, stirring occasionally; add a splash of water if they start to brown too quickly.
Remove the skillet from the heat and spread the cooked onions and garlic evenly across the bottom of the pan.
In a mixing bowl, lightly beat the eggs with a fork or whisk, then stir in the milk, crème fraîche, grated parmesan, ½ teaspoon sea salt, and ½ teaspoon black pepper (adjust seasoning to taste).
Fold the spinach leaves and chopped smoked salmon into the egg mixture until evenly combined.
Pour the egg, spinach, and salmon mixture over the onions in the skillet, spreading it evenly.
Bake in the preheated oven for 20 minutes, then check doneness; continue baking up to 30 minutes total until the edges and surface are lightly browned and the center is set.
Allow the quiche to cool in the skillet for about 10 minutes, then slice into wedges and serve.
