Preheat your oven to 350°F and lightly oil a 9-inch or 9.5-inch pie dish.

In a large mixing bowl, whisk together the eggs, milk, sea salt, and a generous amount of freshly ground black pepper until well combined.

In a large skillet, heat the olive oil over medium heat. Add the sliced shallots, seasoning them with a pinch of salt. Sauté the shallots for about 4 to 5 minutes, stirring occasionally, until they soften and become translucent. Add the broccoli florets along with the water. Continue to cook, stirring frequently, for about 4 minutes, until the water evaporates and the broccoli turns a vibrant green, slightly tender but still crisp.

Arrange the sautéed vegetables evenly in the prepared pie dish. Sprinkle the grated Gruyere cheese over the vegetables, ensuring an even distribution. Pour the egg mixture over the cheese and vegetables, gently tilting the dish if needed to spread the liquid evenly. Finally, garnish with fresh thyme leaves or chopped chives.

Place the dish in the preheated oven and bake for 30 to 40 minutes, or until the quiche is set and the top is lightly golden. Allow the quiche to cool for a few minutes before slicing and serving.