Place one of the oven racks in the upper third of your oven. Preheat it to 350°F.

In a 10-inch nonstick skillet, heat the oil over medium heat. Add the sliced leeks, let cook, stirring occasionally, until softened, about 5 minutes. Add the mushrooms to the skillet, let cook, stirring occasionally, until softened and liquid has evaporated, about 8-10 minutes.

While the vegetables cook, whisk the eggs, heavy cream (or half and half), salt and pepper in a large bowl. Mix in 1/2 cup of the cheese.

Pour the eggs over the mushroom mixture, shaking the pan, if necessary to distribute evenly. Let cook, without stirring, until the edges have started to set, about 5 minutes. Top with the rest of the cheese, place into the oven. Bake until center has just set, about 20-25 minutes.