Preheat the oven to 350°F and grease a 9-inch or 9.5-inch pie dish.

In a large bowl, whisk together the eggs, milk, salt, and several grinds of pepper.

Heat the olive oil in a large skillet over medium heat. Add the shallots and a pinch of salt and cook, stirring occasionally, for 4 to 5 minutes, or until softened. Add the broccoli and water and cook, stirring, for 4 minutes, or until the water has evaporated and the broccoli is bright green and has just lost its raw bite.

Transfer the vegetables to the prepared pie dish and spread in an even layer. Sprinkle the cheese evenly on top. Pour in the egg mixture and gently shake the pan to distribute. Sprinkle with the thyme and bake for 30 to 40 minutes, or until the eggs are set.