Preheat your oven to 350°F and generously grease a 9-inch or 9.5-inch pie dish with olive oil.

In a large mixing bowl, whisk together the eggs, milk, salt, and several grinds of fresh black pepper until well combined.

Heat the olive oil in a large nonstick skillet over medium heat. Add the sliced shallots with a pinch of salt and cook, stirring occasionally, for 4 to 5 minutes until softened and translucent.

Add the broccoli florets and water to the skillet. Cook, stirring frequently, for 4 minutes until the water has completely evaporated and the broccoli is bright green and tender-crisp.

Transfer the cooked vegetables to your prepared pie dish, spreading them in an even layer across the bottom.

Sprinkle the grated Gruyere cheese evenly over the vegetables, then pour the egg mixture over everything. Gently shake the pan to ensure even distribution.

Sprinkle the fresh thyme or chives on top and bake for 30 to 40 minutes, or until the eggs are completely set and the top is lightly golden.