Preheat oven to 350°F and then chop the zucchini, onion, and mushrooms.
Place each type of vegetable into a separate small microwave-safe bowl and then add a teaspoon of butter to each bowl. Microwave the vegetables, one at a time, for one minute and then set aside.
Combine the 5 eggs, flour, baking powder, and salt into a large bowl and thoroughly whisk it all together. Then add the egg mixture to a glass baking pan. Add vegetables to egg mixture and incorporate well. Finally, add in the cheddar cheese and mix it all together.
Bake the quiche for 35-40 minute or until top is golden.