Lightly brush a 10-inch round pan with vegetable oil. Cut a piece of parchment, place in the pan, brush again with oil and set aside.
Sauté onions and garlic in olive oil until soft. Add red pepper, mushrooms and zucchini and continue to cook until all the vegetables are soft. Drain the liquid and add the oregano, salt and pepper. Set aside to cool.
In a large mixer bowl, beat the eggs, milk, Cheddar and Parmesan cheese on low speed. When combined, add cream cheese in small, bite-size bits.
Put cooked spaghetti into baking pan. Add vegetables. Pour in egg mixture and mix with hands to equally distribute ingredients. Pat down so liquid covers solids.
Bake for 30 to 40 minutes until firm to the touch and lightly brown. The frittata will puff and settle when cool. Serve immediately on bed of lightly sautéed tomato sauce.
*If you want to serve frittata the next day, bake for only 25 minutes, cool and then refrigerate. When ready to serve, cut into slices and reheat for 10 minutes at 400°F.