Preheat your oven to 350°F. Lightly grease a 9-inch or 9.5-inch pie dish with olive oil.
In a large bowl, whisk together the eggs, milk, sea salt, and several grinds of black pepper until fully combined. Set aside.
Heat 1 tablespoon of olive oil in a large skillet over medium heat. Add the sliced shallots with a pinch of salt and sauté for 4 to 5 minutes until softened.
Stir in the broccoli florets and ¼ cup of water. Cook for about 4 minutes, stirring occasionally, until the water evaporates and the broccoli turns bright green.
Spread the broccoli and shallots evenly in the greased pie dish.
Sprinkle the grated Gruyere cheese evenly over the vegetables.
Pour the egg mixture over the top, tilting the pan gently to distribute evenly. Sprinkle fresh thyme or chives on top.
Bake the quiche for 30 to 40 minutes, or until the eggs are set and no longer wobbly in the center. Let cool for 5 to 10 minutes before slicing and serving.