Take 6 large eggs from frig to bring to room temperature.
Preheat oven to 350 degrees F.
Wash zucchini. Coarsely grate them. Toss zucchini shreds with 2 t salt and let it rest for 5 to 10 minutes.
Take handfuls, squeezing gently to remove all excess water and pile on paper towels to drain.
Heat the butter and olive oil in a large skillet over medium heat. Add the chopped scallion. Sauté until almost tender. Turn heat to medium high; add zucchini and some freshly ground pepper and sauté for 8 to 10 minutes, stirring almost constantly. Remove from heat.
To serve as a starter, butter 6 small individual casseroles or ramekins. Divide the zucchini among them and make a shallow hollow in the center of each one. (If it is meant as a main dish, use slightly larger oval oven-safe dishes and make two hollows in each.).
15 to 20 minutes before you want to serve, break one egg carefully into each little hollow. Sprinkle the eggs with a little salt and pepper. Bake them for about 15 minutes at 350°F.
The white should be completely set and the yolk still a little soft inside.
If upgrading the dish :-) by using hollandaise or another sauce, spoon sauce over each egg when you take them out of the oven, and serve immediately.